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� " �zG � ���2GH02�9"D$������ � . $�9 � �! � # � G�� � ? I � h k ! # � � �$��%e6 � �h$�9N�d$ $59 � # � � �j" ��G � # ��+0$?()� $5�+&�9" $ # ����$59 �A��%'��-)"D$i� . ��% ! 9N��� � � ! �Z��+��&G;��" �.$�� � � � � $ 0�� # �5-H�M+&���.$m% � " � % �/. �>$ � ��"8QwL�9 � �J�Z��+(2 �?� .G � # �&+6$M+���"D$ # ���X��"P$59 �`��% ! 9N��� � � ! �Z��+��&Gb�9"z� $�� � � � � $ ��" 9 �9" +&��%��0"D$M0�� # �\- �=+&���.$m% � " � % �/. �>$ � ��" � �� �P�$- � $ # �>$���G � � }H��% � " � "H�P$�9H�3%/�j} � %s- % �P�K��G � " � ! � � "D$`0�� # �5- � $59 �3+�- #5# ��"D$?�$�K�2G � " � ! � � "D$ . � #$�9H�gU�r I � - �<� �.$���% (?9 �0" � ! ! � � " �7$59 � �e�,G � +N�&G �e�~} � %s- %�� � �.$��>" +&��$5� ?)���P�Z� ! �\� !H# � � �P�&%�9" G $�9H� �! � # � G�� � ? I � h k ! # � � �$�&% �59 ��(j" � " k � ��- # � / Q �HQ k � # %'���.$?�P�K��G � " �s�P��0����$�h$�9H� # � � �� � $'$5�$�;G �PO � # ��" +�� � " $�9H�s%/�j} � %s- % �$����G � "H� ! � � "D$ . � # $�9 �k$ (`��%��6$�9 �,G �&Q �)-4$f$�9 � �3G �PO � # ��" +��

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0.9 1 1.1 1.2 1.32000

3000

4000

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����� ��� � � � � 0)� �1'*%.+�� � +��.% �� ������B� � � �0):%�+-� �1'*%��4)��C�9+ �4):% �=+�� %�� ���6 � ���@%� �.% ������%� � � �&%�+-� �1'*%��4+�)�' � ����� �4%���+�) �9+-%� %���+�����0):%B'*)��"�6'������ ��� �7+-' ���7)�� �9+ �4):%+ ���� ����� �9+����@�0� �-��'*)��� � + � �����,+-)��C%.+ � +���� %��4+�)�� �=+ � %�� ������%� � �.+��&� ��2�����4)���%�� %��>'!���&%��4+�) ' � �B!������1+�� ����?� � � ������� ���,6 �B!�� ���1+�� ���� '*)��?� +���� � �� � '*�����2��� � ���.%.') � %.+6� +�� �4'�� ������>+�� �� ����� �.� 0�3%��&� ���

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0.9 1 1.1 1.2 1.30.9

1

1.1

1.2

1.3

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λp

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����� ��� � � � � '������2��� �7+-'!���4)�� �,+��7):%6�� ������ �+�����0):%?+� �>'�%��4)�� �9+ �4):% � +�� %�� ���6 � ����%���.%������%� � ���4%� +-� �1'*%��4+�)�' � ����� �4%���+�) �9+-%� %�� �+��� � 0):%�'*)�� �6'������ ��� �4+�'!���4)�� �9+ �4):% � ������� �9+����"�0���-� '*)�� � � +������.�9+-)��$%.+/� +���� %��4+�)����=+���%��)������%��� � +��4���� �����4)�� %��?%��>'!���&%��4+�) ' �� !�� ���>+�� �� ����� � ��� ��� � ��� 6�� !�� ���>+ � ���� '*)��$� +���� � � � '������ ��� �$����% '*) � %.+ � +�� �4'�� �����1+�� ����?������� A� %��4�� � ��

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0 0.2 0.4 0.6 0.8 12450

2500

2550

2600

2650

2700

2750

2800

ω1

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����� ��� � � � � � � �1'*%��4)��<� +��.%����� ������ � ������:%��4)�� �='*�3% + � ��� � +��8%���� �4)�A'!� � + � � �4)�'�%��7+-) �=+��� ��6�4'*%��4+�) '���� ������ %.+C%����� 6������8%���.% ������%��� �=+ � � �'� �%�

G �,�&� # �&�\- �K$ � "|� � ��" � +N+&�9"D$3+&���.$3G �PO � # �0" +&���&QCn . $59 �s� � $��&% � ! � # �d$�� # (]�>" $5�&G �/�\���$-4$ � ��"�( � $59%'� # ����% ! 9 ��� � �A�9"�+����.$A% � " � % �5. �>$ � ��"86O$59 � �� � # � G'� � ? I � h k ()��-H�$G_" �9$ � �h$59 �[� ! ! # � ! # � �>$��$��,���yQ

� ,/+-,/+ �+�� �!��� �� � ���� � � �� �� � 3 � 5 �����

L�9 ��%s- �K$ � I � ��� �&+6$ � 0�� � � " �&� # ?)��% � � " �>$ � ��" . � # %s- �$�>$ � ��"a()���d� ! ! � � ��G $5� � �9$�9 $����.$s� �.$���%���QL�9 �3� O ��+0$?��"/+&��� $?�9" G�� $�� �H� � � $ %/� # � � " . � # G �PO � # ��"D$)02���P- �&�]� . ��" �<� . $�9 �<(`� � ��9D$ � " � . ��+6$�� # ��>$<�/� � 0���"e0��2�$- �;� . % $ . � # $�9 �gU�r I � -H�E� � $���% � # �b�59H�j(j" � " k � ��- # �&� / QPY ���9" G / QPY�Y�QE:E�f$�9H�. ��+6$�� # ��� � � � " + # �&���5��G86m%'� # �d�&% ! 9N��� � � � � ! �$��+&��G1��" � ! � # �>$ � " ��+&��� $��z�9" G �$���5�;��"1� $�� �H� � � $ %�� # � � "86 �2���R} ! ��+0$5�&G8Q

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0.2 0.4 0.6 0.8 10.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

ω1

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����� ��� � ��� � � '������ ��� �4+�'!���4)�� � � ����� � �����:%��4)�� � '�3%.+�� � � �=+���%�� � �7)�0' ����+ � ���7) '�%��4+�)�=+ ��2� �7'�%��7+-) '���� ������ %.+ % ������6 � ���8%���.% �����.%��� �=+ � � � � ��

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0.8 1 1.2 1.4 1.61000

2000

3000

4000

5000

6000

7000

8000

9000

λp

cost

OPF Max. Dist ω

1 = 1e−4

ω1 = 2e−4

ω1 = 4e−4

ω1 = 10e−4

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k � ��- # ��� / QuY��|�9" G / QPY43aGH� ! � +0$k$59 � # ���5- �W$��d� � $�� � " �&G .7# ��% � ! ! � � " ��$59 � � � " �&� # ?)��% � � I"N�>$ � ��" . � # %s-H�Z�>$ � ��"86 $�9H� �|�~} � %s-H% � � � $��9"H+&��$5� ?)���$�$� ! �5�s�9" G $�9 �k$ # ��G � $ � ��"N��� � h k $�� $�9H�U�r I � - ��� � $���%eQ�:E�A� } ! ��+0$���G86n$59 �?�5���P-4$ � ��" �=� � $�� � " �&G .7# ��% $59 ��� � " �4� # ?)��% � � "N�>$ � ��" � . � # %s- I�Z�d$ � �9"d� # � � � �������� � $�9 �?�5���$-4$ � ��" �M� � $�� � " ��G .7# ��% $59 ���|�~} � %s- % � � �.$��>" +&�M$5�2?)���$�$� ! �5�?�>" G" � # %/�2� � h k QS: $<�$��()� # 02���P- �&�[� . � � 6C$59 � � � " �4� # ?)��% � � "N�d$ � �9"e�\���$-4$ � ��"H�j$5��" G�$������7$5�7$�9H��|�~} � %s- % � � �.$��>" +&�]$�� ?)���$�$� ! �\�C�5���P-4$ � ��" �&6 (?9H� # �&���3�d$E9 � ��9 � # 02���P- �&�?� . ��� $59 �&�\�b�5���$-4$ � ��" �� ! !H# �K��+�9 $�9H� � h k �5���P-4$ � ��" ��Q�:<� � �P�$- � $ # �d$��&G � " k � ��- # � / QuY / 6n$�9 �gY�Y � I � - �?� � $���% �R}^9 ���H� $���G� � % � �Z� # +�9N� # ��+0$5� # � �.$ � +��E�2�j$59 �zU�r I � - ��� � $��&%�Q

L�9 �?G � �5��GH02�9"D$��2����� . $�9 ��� � " �4� # ?)��% � � "N�>$ � ��" � . � # %s-H�Z�>$ � ��" � � � �$�$-H�.$ # �>$���G � " k � ��- # � / QuY43HQ� � �5� # 0��M$�9N�d$i$59 � # � � �m�E�P�K��G � " � ! � � "D$46~(?9 � +�9b02� # � ���i( � $�9S$�9 �A0��2�$- ���m� . $�9 �A()� � ��9D$ � " � . �2+0$�� # �&6(?9 � # �,$�9 �����$��� # � $�9 % �5���P-4$ � ��" � ��� � +4�2�$� �\( � $�+�9 ���b��0�� #;.7# ��% %��~} � % �/.�� " ��� $�� �H� � � $ %�� # � � " �

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0.8 1 1.2 1.4 1.60.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

λp

λ *

OPF Max. Dist ω

1 = 1e−4

ω1 = 2e−4

ω1 = 4e−4

ω1 = 10e−4

����� ��� � � � � � '������2��� �7+-'!���4)��<�,+��7):%��� ������8�+��� � 0):%@+� �>'�%��4)��<�9+ �4):% �=+ �@%�� � �7)�0' � � +�� � � 6)�'�%��7+-)��@'*)��2� '������2��� �$����% '*) � %.+�� +�� �7'����� �=+ ��2� �7'�%��7+-)�� �=+��B%����� 6������8%���.% ������%���"

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0.8 1 1.2 1.4 1.60

1000

2000

3000

4000

5000

λp

cost

ω1 = 1e−4

ω1 = 3e−4

ω1 =10e−4

OPF

����� ��� � � � � �@� �>'�%��4)��2� +��.%����� ������ � � � �0):%"+� �>'�%��4)����9+ �4):% �=+��C%�� � �4)�0' ����+ � ���7) '�%��4+�)'*)�� %��>'!���&%��4+�) ' � �B!�� �=+ ��2� �7'�%��7+-)�� � +��B% �� ��� ��6�������%����% ������%� �)

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0.8 1 1.2 1.4 1.60

1000

2000

3000

4000

5000

6000

7000

λp

cost

Fixed Stability Margin VSC−OPFOPF

����� ��� � � � � +��.%�� � �����8� � � �0):%B+� �>'�%��4)����,+��7):% �=+�� %�� ���&���� � % ' ��� ���4%�� � ' ����7) ')�� %��1' ��� 6%��7+-)�' � � !�� � +��� � �4'*%��4+�)�� �=+��B%�� ����� 6 � ���8%� �.%������.%���"

$��g% � " � % �5.&� " � +&��� $��&QL�9 � � � � ! � # $�� /# �&�$�>$���G $5�k$�9 � G �uO � # �0" +&� � �0$ (`�&�0" % � �9"HG % $ � ��+&��% � "H��5%����P�$� # �>$/9 � ��9 � # �$�K��G � " � �$�R0��&�P� v $59 � # � . � # ��6j$�9 �|�&% ! 9N��� � �/� . $�9 � �7$�� # % � " $�9 �a� ��� �&+6$ � 0��. -)" +6$ � ��"�G �&+ # �4�2�5�&��Q

� ,/+-, � � �� � � 3���� ��� ����� �� ���'� �� � 3 � 5 �����

L�9 � $�9 � # G/�\�0$)� . "X- %'� # � +4���w�9" ��� � � � � "K0����u0������ ! ! � � " �S$�9 � k � }H��G � $�� � � � � $ �|� # � � " . � # %s-H�Z� I$ � ��",$�� � �9$597$����.$)� �.$5�&%'�&QA>[��+4���P�)$�9N�d$�$59 � �A%��6$�9 �,G � � � ��� � +4���P� �>" � h k (?9 � # �3�;% � " � %b- %�.$�� � � � � $ %�� # � � " � �?�0" �5- # �&G8Q

k � # � �9$59 $����.$*� � $���%���6E�1% � " � %s-H% �.$�� � � � � $ %�� # � � " � % ��� � � � . Y ! QV-8Q � �/-H�5�&GcQ n#"���0" � # ��� 6d$�9 �`���$��� # � $�9 % . ��-)" Gd�[�5���P-4$ � ��"S$�9N�>$m��"H�5- # �&Gk$�9 � �i+&��" � $ # � � "D$ v 9 ��()�R0�� # 6�$�9 � � # �&�\- �K$5�&G

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� "C9 � ��9 � # � ! � # �>$ � " ��+&��� $���QM:f+&��% ! � # � �\��"3� . $�9H��� ! � # �>$ � " �)+&��� $���� . $�9 �?Y�Y � I � -H�8� �.$5�&% 0�� # �\- �+&- #5# �0"D$?�$�K��G � " � ! � � "D$ . � # $59 � k � }^�&G � $�� � � � � $ �|� # � � "*�9" G $ # ��G � $ � ��"N��� � h k . � # %s-H�Z�>$ � ��" � � ��59H�j(j" � " k � ��- # � / QuYjU^Q � � �5� # 0�� $�9N�>$�$59 �EG �PO � # �0" +&� � ",$�9H�j$5�9$��2�N� ! � # �>$ � " � +����.$5� � " + # �4���\�&�]�2�$�9H�C�$�K��G � " � � " + # �4���\�&��6NG - �:$��V$59 � . ��+0$h$59N�>$?�d% � " � %s- % �.$�� � � � � $ %�� # � � " � � � � � " �d�0" . � # +&�&Gc6(?9 � +�9 � �&+���%����[�gGH��% � "N�9"D$?+&�9" �.$ # � � " $[�2�j$59 � � �.$5�&% ���6$��[+&�P���5� # $��g+&���P�Z� ! �5��Q

� ,/+-,10 � ��� ���' ����'� � � �� � �83 � 5 �����

L�9 ��" �R}Q$M�5�0$m� . ",-H%�� # � +&���X�9"N�2� � � � � "K0����u0����=� ! ! � � " �h$59 � _ �K����h # ��� # ��%�% � "H� . � # %s-H�Z�>$ � ��"f$5�� �9$�9V$����.$� � $���%���Q>[��+4���P�n$�9 �>$ $�9 � � G �4� � ��$�� G � +=" �[�3�.$�� � � � � $ %�� # � � "�$59N�>$ � ��" �9$A� � � " G � "H�+&�9" �.$ # � � " $&6 � -4$�$�9N�>$ � . 0 � ���Z�>$5�&G86 � " + # �&���5���k$�9 ��� ��� �&+6$ � 0�� . -)" +6$ � ��""+&���.$4Q|:<� � �$�$-H�.$ # �>$���G � "k � ��- # � / QuY�WH6 $59 � _ �K���Eh # ��� # ��%'% � " � . � # %s-H�Z�>$ � ��" �59 � . $���G $�9H� � % ! � # $��>" +&��� . +����.$��2� � �� "H+ # �&���5��G8Q�:<��$�9 �3�P�K��G � " �;�$��0��&� � � � " + # �4���\�&G86Q$�9 �<+&��� $�� . %�� � "D$�� � " � " �]$�9 �EG �&� � # ��G*% � " � %b- %�.$�� � � � � $ %/� # � � " � " + # �4���\�&�&6c�9" G86w��0���"D$�- ���$� 6 . � # +&��" � $��9"D$302���P- �&�[� . �2�$�8(`� � ��9 $ � " � . ��+0$5� # � �$�9H�3% � " � %b- % �.$�� � � � � $ %�� # � � " � � # �&G -H+&�&G $5� . � # � Q � 9 �0"*�$�&�\�)(`� � ��9 $ � � ! �$��+&��G���"*+����.$?�>" G� # �4�>$5� # ()� � ��9D$ � � ! �$��+&��Ge�9"e� $�� � � � � $ 6 � QT�2QP6 . � # �\%/�2�$�$� # 02���$-H�&�?� . � � 6)$�9 �;% � " � %b- % � $�� �H� � � $ %�� # � � " � �]%/� � "D$�� � " �&Gc6H�����\9 ��(j" � " k � ��- # � / QPY�r^Q�n�$ � � . ��-)" G � "7$�9 �]"X- %'� # � +4�2�8�9"N�2� � � �M$�9N�d$� "

G �^��� " �9$� # �4�>$5� � O �&+6$�$59 ���5���$-4$ � ��"d� . $59 � ! # � � �$�&%�6�(?9 � +�9 � � $�� � ���R} ! �&+0$5�&G86,� � " +&� $59 � # �� � "H� � ��" � +)$ � "g9N�40 � " �;� �.$�� � � � � $ %/� # � � "d� # �4�d$�� # $59N�9"V$�9 �[G ��� � # �&Gg02���$- �;@ $�9 � � $���"HG � $��z���$�\�# ���5- �W$ � "*� # �4�>$5� # � ! � # �d$ � "H�d+&��� $���D Q

� � % � �$� #A# �&�\- �K$5��()� # �;� � $�� � " ��G . � # $�9 �gY�Y � I � - �?� � $���%eQ

� ,/+-,/. �83 � 5 ����� � ����� �!� � � �� � � ���� �-�������!�� �

L�9 � +=" ���8�5�6$�� . "X- %'� # � +4���c�9"N��� � � � � "K0����P0����?� ! ! � � " ��$59 � _ �K��� h # ��� # ��%�% � "H� . � # %s- �Z�d$ � �9"86(?9 � +�9 � � ! # � � � � � $59 � � ���.$?+���% !H# ��% � �\� . � # �b� � ? I � h k 6 +���" � � G � # � "H� # �4�2+0$ � 0�� ! ��()� # +����.$5�$�� � �>$�9�$5�&� $3� � $��&%'�&Q[n#"�$59 � �[+4�2�5��6C$�9 � � GH�4� � �h$����2G G � ! ��" ���K$ $5�7$�9H�z� � � �&+0$ � 0�� . -)" +6$ � ��"� . ����" � # �>$5� # �C� # �k" �>$3� ! � # �>$���G �>$:$�9H�sG �&� � # �&G ! �j(`� #3. �2+0$�� # Q k � ��- # ��� / QuY ���>" G / QuY �/G � ! � +6$$�9H� # �&�5-H�K$��;� � $�� � "H�&G . � # � ! � # �d$ � "H��+����.$b�9" G1%/�j} � %s- % �$�K��G � " � ! � � "D$&6m�>" G � # �'�5��%'��(?9N�d$

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0.8 1 1.2 1.4

2000

3000

4000

5000

6000

7000

8000

9000

λp

cost

OPF Max. Dist ω

1 = 1e−4

ω1 = 2e−4

ω1 = 4e−4

ω1 = 10e−4

����� ��� � � � � � + ��%� ������ �+��� � 0):% +� �>'�%��4)��?�9+ �4):% �=+ � %�� � +�' � !4�>+ � �>'!� � �4)�� � � � � �%� �;��� ���� ����� ��� ')��?%��>'!���4%��7+-)�' � � !�� �=+��� � �4'�%��4+�)�� �=+ � %������6 � ��� %����%������.%���"

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0.8 1 1.2 1.4

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

λp

λ *

OPF Max. Dist ω

1 = 1e−4

ω1 = 2e−4

ω1 = 4e−4

ω1 = 10e−4

����� ��� � � � � � '������ ��� �4+�'!���7)�� �9+ �4):%�� � ����� � � � �0):% +-� �>'�%��7)�� �9+��7):% �=+ � % �� � +�' � !4�>+ 6���1' � � �4)�� � � � ���%� �;��� � � � � � � � � � %��>'!���&%��4+�) ' � �B!�� '*)�� � '������ ��� � ���.%.'*)��+�%.+�� +�� �4'�� ���=+ ��2� �7'�%��7+-)�� � +��B% ������6 � ���8%���.% �����.%���"

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0.8 1 1.2 1.4 1.61000

2000

3000

4000

5000

6000

7000

8000

9000

λp

cost

OPF Max. Dist ω

1 = 1e−4

ω1 = 3e−4

ω1 = 5e−4

����� ��� � ��� � � + ��%� ������ �+��� � 0):% +� �>'�%��4)�� �9+��7):% �=+ � %�� � +�' � !4�>+ � �>'!� � �4)�� �=+�� �2� �4'�%��4+�)� +�)������� ��4)�� � 0'*� %��4����9+0� ��� + �.%�� � � � � �%� �;� � � � � � � � � � � � � � ��� ���%� �� '*)��"%��1' ���4%��4+�)�' � � !���=+ ��2� �7'�%��7+-) �=+ � %������6 � ��� %����% ������%� �)

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0.8 1 1.2 1.4 1.60.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

λp

λ *

OPF Max. Dist ω

1 = 1e−4

ω1 = 3e−4

ω1 = 5e−4

����� ��� � � � � � '������2��� �4+�'!���4)�� �9+ �4):% � ������ � � � �0):% +� �>'�%��4)�� �9+��7):% �=+�� � +�' � !4�1+ ���1' � � �4)���=+ ��2� �7'�%��7+-) � +�)������� ��4)�� �0'*� %��&� �9+0� ��� +��.%�� � � � � �%� �;��� � � � � � � � � � � � � � � � �%� �� �%��>'!���&%��4+�) ' � �B!��6'*)��<� '������ ��� � ���.%.'*)��+ % +B��+�� �7'�� �� �=+�� �2� �4'�%��4+�)�� �=+ � % �� ���6 � ��� %���.% �����.%���"

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z�� K�D!8:PQD�8 E � < �7G �:476 Z ~ � K�D � �:8:D�@?@698 < � �XD �X4N6 B E 8 < D LN> ~ Z � ! } & P78:D �7@69O > 8J6 >:L @ �X69A < �>:< O < @ E 8 KF4 E 8 E K��X698 <> � < K >CE!> D�� > 698JB!6�A < � Z 6�K � < D � ?N� � � Y HT6 �NK96�P � �:4 E � LN>:< �NG �X476 Z ~ � O�D`A76�@8:6 >:L @ � >_< � 4 <G 4N6�8 O E � < O L O @?D E A < �7G @69B!6�@ >UE�� AW8:69A L K�69ACK�D > � > H E!> 6���PQ6�K �:6�A � & <?G�L 8:6 ��� � � <>_EP7@?D��|D!R �:476 8:6�A L K � < D ��< � �X4N6�:D�� E @�D!P]698 E � < �7G K�D > � > ��� < � K9D�8:PQD�8 E � < �7G �X476 Z ~ � < � �XD �X476 & < �N69AZ � E � < @ < � � �\E 8 G!< � ~ Z � ! } & R�D�8:O L @ E � < D � �/Z 476;A < � 6�8J6 � K96 < � �X4N6�:D�� E @"D�PQ6�8 E � < �7G K�D > � > 354N6 �> D!@?B < �7G �:476 �X8 E A < � < D �7E @ } & P78JD��7@69O <?>$< � K9@ L A769A < � & <?G�L 8:6 ��� � � �:D�A76�O�D �7> �X8 E �X6 �X4 E �$35476 � HK�D �7>:< A76�8 < �7G �QD��X4 > � E � < @ < � � E�� A D�PQ6�8 E � < �7G K�D > � > H �X476 �Q6 � 6 � � > D!R < � K9D�8:PQD�8 E � < �7G �X4N6 Z ~ � E 8J6

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