
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 intervals of monotonicity for the function $f(x)$;
- $(2)$ If for any $x \in \left[0, \frac{\pi}{2}\right]$, $f(x) \geqslant kx$ always holds, find the range of the real number $k$;
- $(3)$ Let the function $F(x)=f(x)+e^{x}\cos x$, where $x \in \left[-\frac{2015\pi}{2}, \frac{2017\pi}{2}\right]$. Draw all the tangent lines to the graph of the function $F(x)$ at point $M\left(\frac{\pi-1}{2},0\right)$, and let the abscissas of the points of tangency form the sequence $\{x_n\}$. Find the sum $S$ of all terms of the sequence $\{x_n\}$.

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