
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 $f(x)=m\sin 2x-\cos^2x-\frac{1}{2}$, where $x\in\mathbb{R}$, and $\tan \alpha=2\sqrt{3}$, with $f(\alpha)=-\frac{3}{26}$.

$(1)$ Find the value of the real number $m$ and the smallest positive period of the function $f(x)$;

$(2)$ Find the intervals of increase for the function $f(x)$ on $[0,\pi]$.

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