
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 $f(x)=\sin(2x+\frac{5π}{6})-\cos^2x+1$.  
$(1)$ Find the minimum value and the interval of monotonic increase of the function $f(x)$.  
$(2)$ Let angles $A$, $B$, and $C$ be the three interior angles of $\triangle ABC$. If $\cos B=\frac{1}{3}$ and $f(\frac{C}{2})=-\frac{1}{4}$, find $\sin A$.

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