
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 $f(x)=\sin x$, $g(x)=x-m$, where $m\in \mathbb{R}$.  
$(1)$ Find the area of the triangle enclosed by the tangent line to the curve $y=f(x)$ at the point $\left(\frac{\pi}{3}, \frac{\sqrt{3}}{2}\right)$ and the two coordinate axes.  
$(2)$ Let $h(x)=f(x)\cdot g(x)$, $x\in \left(-\frac{\pi}{2}, +\infty\right)$. Let $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ (where $x_{n} < x_{n+1}$, $n\in \mathbb{N}^{*}$) be the extreme points of $h(x)$. If $h(x_{1})+h(x_{2})=0$, find the value of $m$.

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