
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 $f(x) = a\sin x + \cos x$ whose graph passes through the point $\left( \frac{\pi}{2}, -1 \right)$.  
(1) Find the smallest positive period of the function $f(x)$ and its intervals of monotonic increase.  
(2) If $\theta \in \left(0, \frac{\pi}{2}\right)$ and $f(\theta) = \frac{1}{2}$, find the value of $\sin 2\theta$.

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