
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 $f(x)=\sin x$, $g(x)=e^{x}\cdot f'(x)$, where $e$ is the base of the natural logarithm.  
(I) Find the equation of the tangent line to the curve $y=g(x)$ at the point $(0,g(0))$;  
(II) If for any $x\in\left[- \frac {\pi}{2},0\right]$, the inequality $g(x)\geqslant x\cdot f(x)+m$ always holds, find the range of the real number $m$;  
(III) Investigate the number of solutions to the equation $g(x)=x\cdot f(x)$ when $x\in\left[- \frac {\pi}{2}, \frac {\pi}{2}\right]$, and explain the reason.

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