
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
Let $f(x)=\sin(2x+\frac{\pi}{6})-2\cos^2x$.

1. Find the interval of monotonicity of $f(x)$.
2. In triangle $ABC$, if $f(\frac{A}{2}-\frac{\pi}{6})=-\frac{5}{4}$, $\overrightarrow{CD}=2\overrightarrow{DA}$, $BD=\sqrt{10}$, $\cos \angle ABD=\frac{\sqrt{10}}{4}$, find the value of $BC$.

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