
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 $\overrightarrow{a}=(\sqrt{3}\cos \omega x,\sin \omega x),$ and $\overrightarrow{b}=(\sin \omega x,0),$ where $\omega > 0,$ let the function $f(x)=(\overrightarrow{a}+\overrightarrow{b})\cdot \overrightarrow{b}-\frac{1}{2}.$ If the graph of the function $f(x)$ is tangent to the line $y=m$ (where $m$ is a constant), and the x-coordinates of the tangent points form an arithmetic sequence with a common difference of $\pi$:

(1) Find the expression for $f(x)$ and the value of $m$.
(2) Translate the graph of $y=f(x)$ to the left by $\frac{\pi}{12}$ to obtain the graph of $y=g(x)$. When $x\in (\frac{\pi}{2},\frac{7\pi}{4})$, the x-coordinates of the intersection points of $y=g(x)$ and $y=\cos \alpha$ form a geometric sequence. Find the value of obtuse angle $\alpha$.

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