
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
Given the equation $\frac{{\sin x}}{x}=({1-\frac{{{x^2}}}{{\pi^2}}})({1-\frac{{{x^2}}}{{4\pi^2}}})({1-\frac{{{x^2}}}{{9\pi^2}}})\cdots({1-\frac{{{x^2}}}{{n^2\pi^2}}})\cdots$ and the Taylor series expansion $\sin x=x-\frac{{{x^3}}}{3!}+\frac{{{x^5}}}{5!}+\cdots+\frac{{{{(-1)}^{n-1}}{x^{2n-1}}}}{({2n-1})!}+\cdots$, find the value of $\frac{1}{{1^2}}+\frac{1}{{2^2}}+\frac{1}{{3^2}}+\cdots+\frac{1}{{n^2}}+\cdots$.

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