
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 $a=(\cos ωx,\sin ωx)$, $b=(2\cos ωx+\sin ωx,\cos ωx)$, $x\in R$, $ω > 0$, let $f(x)=a\cdot b$, and the smallest positive period of this function is $\frac{π}{4}$.

(1) Find the value of $ω$;

(2) Find the maximum value of the function $f(x)$, and the set of $x$ for which $f(x)$ attains this maximum value.

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