
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 vectors $\overrightarrow{a}=({\sqrt{3}\sin x,\cos x}),\overrightarrow{b}=({\cos x,\cos x})$, and the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}-\frac{1}{2}$.  
$(1)$ Find the interval where $f(x)$ is monotonically increasing.  
$(2)$ If $f({\frac{α}{2}})=-\frac{3}{5},α∈({-\frac{π}{2},0})$, find $\sin \alpha$.

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