
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 $t(x)=2\sin\omega x\cos\omega x+2\sqrt{3}\cos^2\omega x$ $(\omega>0)$, and the minimum positive period of $f(x)$ is $\pi$.  
$(1)$ Find the value of $\omega$ and the interval of monotonic decreasing of the function $f(x)$;  
$(2)$ The graph of the function $f(x)$ is shifted to the right by $\frac{\pi}{6}$ units to obtain the graph of the function $g(x)$. Find the maximum value of the function $g(x)$ when $x\in[0,\frac{\pi}{2}]$.

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