
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 $f(x)=2\sin(\omega x+\phi)$ where ${\omega>0, |\phi|<\frac{\pi}{2}}$ is monotonically increasing on the interval $(-\frac{\pi}{8},\frac{3\pi}{8})$, and satisfies $f(0)=-\sqrt{2}$, and for any $x\in \mathbb{R}$, $|f(x)|\leq f(\frac{3\pi}{8})$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ Let $g(x)=f(x)\cos(2x)$, find the interval on which $g(x)$ is monotonically increasing on $(0,\frac{\pi}{2})$.

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