
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 $f(x)=e^{x}\sin x$.

(1) Find the monotonic intervals of the function $f(x)$;

(2) If for any $x\in[0, \frac{\pi}{2}]$, $f(x)\geqslant kx$ always holds, find the range of values for the real number $k$;

(3) Let the function $F(x)=f(x)+e^{x}\cos x$, $x\in[- \frac{2011\pi}{2}, \frac{2013\pi}{2}] $. Through the point $M( \frac{\pi-1}{2} ,0)$, draw all the tangent lines to the graph of function $F(x)$, and let the horizontal coordinates of each tangent point form a sequence $\{x_{n}\}$, find the value of the sum $S$ of all items in the sequence $\{x_{n}\}$.

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