
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 $f(x)=\sqrt{13}\cos \omega x\cdot \cos (\omega x-\varphi )-\sin ^{2}\omega x$ where $(\omega > 0)$, $\varphi \in (0,\frac{π}{2})$, and $\tan \varphi =2\sqrt{3}$, and the smallest positive period of $f(x)$ is $\pi$. Find:<br/>
$(1)$ The analytical expression of $f(x)$;<br/>
$(2)$ The range of the function $y=f(x)$ on the interval $[\frac{π}{12},\varphi]$.

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