
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 function $\sin x=x- \frac{x^{3}}{3!}+ \frac{x^{5}}{5!}- \frac{x^{7}}{7!}+…+(-1)^{n-1} \frac{x^{2n-1}}{(2n-1)!}+…$, which has infinite roots due to $\sin x=0$; $(0,\pm \pi ,\pm 2\pi ,\pm 3\pi ,\cdots )$. It can be deduced that: $\sin x=x\left( 1-\frac{x^{2}}{\pi ^{2}} \right)\left( 1-\frac{x^{2}}{4\pi ^{2}} \right)\left( 1-\frac{x^{2}}{9\pi ^{2}} \right)\ldots $. Upon expanding the right side of this equation, we find that the coefficient of $-x^{3}$ is $\frac{1}{\pi ^{2}}+\frac{1}{(2\pi )^{2}}+\frac{1}{(3\pi )^{2}}+… =\frac{1}{3!}$,

i.e., $\frac{1}{1^{2}}+\frac{1}{(2 )^{2}}+\frac{1}{(3 )^{2}}+… =\frac{\pi ^{2}}{6}$. Now, use the function $\cos x=1- \frac{x^{2}}{2!}+ \frac{x^{4}}{4!}- \frac{x^{6}}{6!}+…+(-1)^{n-1} \frac{x^{2\left(n-1\right)}}{2\left(n-1\right)!}$ to derive a similar conclusion.

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