
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 $\overrightarrow{m}=(\sqrt{3}\cos x,-\cos x)$ and $\overrightarrow{n}=(\cos (x-\frac{π}{2}),\cos x)$, satisfying the function $f\left(x\right)=\overrightarrow{m}\cdot \overrightarrow{n}+\frac{1}{2}$.  
$(1)$ Find the interval on which $f\left(x\right)$ is monotonically increasing on $[0,\frac{π}{2}]$.  
$(2)$ If $f\left(\alpha \right)=\frac{5}{13}$, where $\alpha \in [0,\frac{π}{4}]$, find the value of $\cos 2\alpha$.

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