
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]


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Given vectors $\overrightarrow{a} = (\sin x, -\frac{1}{2})$ and $\overrightarrow{b} = (\cos x, \cos(2x + \frac{\pi}{6}))$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$:

(1) Find the interval of monotonic increase for the function $f(x)$.

(2) If the function $f(x)$ has local maximum points on the right side of the y-axis, which form an increasing sequence $\{a_n\}$, find the sum of the first $n$ terms, $T_n$, of the sequence $\{\frac{\pi^2}{a_n a_{n+1}}\}$.

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