
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 $f(x)=\sin \left(2x+\dfrac{\pi }{6}\right)-2\cos ^{2}x$.
(1) Find the interval where $f(x)$ is monotonically increasing.
(2) In $\triangle ABC$, if $f\left(\dfrac{A}{2}-\dfrac{\pi }{6}\right)=-\dfrac{5}{4}$, and $\overrightarrow {CD}=2\overrightarrow {DA}$, $BD=\sqrt {10}$, $\cos \angle ABD=\dfrac{\sqrt {10}}{4}$, find the value of $BC$.

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