
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 the function $f(x) = A\sin(\omega x + \varphi) + 1 (A > 0, \omega > 0, -\frac{\pi}{2} \leqslant \varphi \leqslant \frac{\pi}{2})$ whose graph is symmetric about the line $x = \frac{\pi}{3}$, has a maximum value of $3$, and the distance between two adjacent highest points on the graph is $\pi$.
1. Find the smallest positive period of $f(x)$;
2. Find the analytic expression of the function $f(x)$;
3. If $f(\frac{\theta}{2} + \frac{\pi}{3}) = \frac{7}{5}$, find $\sin \theta$.

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