
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 $f_{1}(x)=f(x)$, $f_{2}(x)=f'_{1}(x)$, $f_{3}(x)=f'_{2}(x)$, $\cdots$, $f_{n+1}(x)=f_{n}'(x)$, $n\in N^{*}$. For example, if $f_{1}(x)=\sin x$, then $f_{2}(x)=\cos x$, $f_{3}(x)=-\sin x$, $f_{4}(x)=-\cos x$, $\cdots$. If $f(x)=e^{\cos x}+e^{-\cos x}$, then $f_{4}(2\pi)=\_\_\_\_\_\_$.

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