
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 functions $f(x)=e^x\sin x-\cos x$ and $g(x)=x\cos x- \sqrt{2}e^x$, where $e$ is the natural constant.

$(1)$ Determine the number of zeros of the function $y=f(x)$ in the interval $\left(0, \frac{\pi}{2}\right)$, and explain the reason;

$(2)$ For every $x_1 \in \left[0, \frac{\pi}{2}\right]$, there exists $x_2 \in \left[0, \frac{\pi}{2}\right]$ such that the inequality $f(x_1)+g(x_2) \geqslant m$ holds. Find the range of the real number $m$.

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