
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 $\overrightarrow{a}=(\cos x, \sqrt{3}\cos x)$ and $\overrightarrow{b}=(\cos x, \sin x)$.<br/>$(1)$ If $\overrightarrow{a}$<span class="special-symbol">∥</span>$\overrightarrow{b}$, $x∈[{0,\frac{π}{2}}]$, find the value of $x$;<br/>$(2)$ If $f\left(x\right)=\overrightarrow{a}\cdot \overrightarrow{b}$, $x∈[{0,\frac{π}{2}}]$, find the maximum value of $f\left(x\right)$ and the corresponding value of $x$.

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