
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 $\overrightarrow{a}=(-\sqrt{3}\sin \omega x,\cos \omega x)$, $\overrightarrow{b}=(\cos \omega x,\cos \omega x)$ $(\omega > 0)$, let the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, and the smallest positive period of $f(x)$ is $\pi$.
$(1)$ Find the value of $\omega$;
$(2)$ Find the intervals of monotonicity for $f(x)$.

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