
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 the vector $\vec{m}=(2\cos x,\sqrt{3}\sin x)$ and $\vec{n}=(\sin x,-2\sin x)$. Denote $f(x)=\vec{m}\cdot\vec{n}$.

$(1)$ Find the interval of monotonic decrease for the function $f(x)$;

$(2)$ Find the range of the function $f(x)$ on the interval $\left[-\frac{\pi}{3},\frac{\pi}{6}\right]$.

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