
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 $\overrightarrow{a}=(\sin \omega x,\cos \omega x)$, $\overrightarrow{b}=(\sin \omega x+2\cos \omega x,\cos \omega x)$, $x\in\mathbb{R}$, $\omega > 0$, let $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$ and the smallest positive period of this function is $\frac{\pi}{4}$.
$(1)$ Find the value of $\omega$;
$(2)$ Find the maximum value of the function $f(x)$, and the set of $x$ for which $f(x)$ attains its maximum value.

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