
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: $\overrightarrow{a}=(\sin x+\cos x,2\sin (x+\frac{\pi }{4}))$, $\overrightarrow{b}=(\sin x,\sin (\frac{\pi }{4}-x))$, and $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$.

(1) If $\tan \alpha=2$, find the value of $f(\alpha)$;

(2) If $x\in[\frac{\pi}{12}, \frac{\pi}{2}]$, find the range of values of $f(x)$.

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