
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Observe the following equations:

\\({{\\left( {\\sin }\\dfrac{{ }\\!\\!\\pi\\!\\!{ }}{3} \\right)}^{-2}}+{{\\left( {\\sin }\\dfrac{2{ }\\!\\!\\pi\\!\\!{ }}{3} \\right)}^{-2}}=\\dfrac{4}{3}\\times 1\\times 2;\\)

\\({{\\left( {\\sin }\\dfrac{{ }\\!\\!\\pi\\!\\!{ }}{5} \\right)}^{-2}}+{{\\left( {\\sin }\\dfrac{2{ }\\!\\!\\pi\\!\\!{ }}{5} \\right)}^{-2}}+{{\\left( {\\sin }\\dfrac{3{ }\\!\\!\\pi\\!\\!{ }}{5} \\right)}^{-2}}+{{\\left( {\\sin }\\dfrac{4{ }\\!\\!\\pi\\!\\!{ }}{5} \\right)}^{-2}}=\\dfrac{4}{3}\\times 2\\times 3;\\)

\\({{\\left( {\\sin }\\dfrac{{ }\\!\\!\\pi\\!\\!{ }}{7} \\right)}^{-2}}+{{\\left( {\\sin }\\dfrac{2{ }\\!\\!\\pi\\!\\!{ }}{7} \\right)}^{-2}}+{{\\left( {\\sin }\\dfrac{3{ }\\!\\!\\pi\\!\\!{ }}{7} \\right)}^{-2}}+\\cdots +{{\\left( {\\sin }\\dfrac{6{ }\\!\\!\\pi\\!\\!{ }}{7} \\right)}^{-2}}=\\dfrac{4}{3}\\times 3\\times 4;\\)

\\({{\\left( {\\sin }\\dfrac{{ }\\!\\!\\pi\\!\\!{ }}{9} \\right)}^{-2}}+{{\\left( {\\sin }\\dfrac{2{ }\\!\\!\\pi\\!\\!{ }}{9} \\right)}^{-2}}+{{\\left( {\\sin }\\dfrac{3{ }\\!\\!\\pi\\!\\!{ }}{9} \\right)}^{-2}}+\\cdots +{{\\left( {\\sin }\\dfrac{8{ }\\!\\!\\pi\\!\\!{ }}{9} \\right)}^{-2}}=\\dfrac{4}{3}\\times 4\\times 5;……\\)

Following this pattern,

\\({{\\left( {\\sin }\\dfrac{{ }\\!\\!\\pi\\!\\!{ }}{2{n}+1} \\right)}^{-2}}+{{\\left( {\\sin }\\dfrac{2{ }\\!\\!\\pi\\!\\!{ }}{2{n}+1} \\right)}^{-2}}+{{\\left( {\\sin }\\dfrac{3{ }\\!\\!\\pi\\!\\!{ }}{2{n}+1} \\right)}^{-2}}+\\cdots +{{\\left( {\\sin }\\dfrac{2{n }\\!\\!\\pi\\!\\!{ }}{2{n}+1} \\right)}^{-2}}=\\)\_\_\_\_．

Present the answer in LaTex format: \boxed{Your answer}
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