
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

The equation of the function is \( y = 2 \cos(\omega x + \theta) \) where \( x \in \mathbf{R} \), \( \omega > 0 \), and \( 0 \leq \theta \leq \frac{\pi}{2} \). The graph of this function intersects the y-axis at the point \( (0, \sqrt{3}) \) and the minimum positive period of the function is \( \pi \).

1. Find the values of \(\theta\) and \(\omega\).
2. Given the point \( A \left( \frac{\pi}{2}, 0 \right) \) and a point \( P \) on the function's graph, with point \( Q \left( x_0, y_0 \right) \) being the midpoint of segment \( PA \). When \( y_0 = \frac{\sqrt{3}}{2} \) and \( x_0 \in \left[ \frac{\pi}{2}, \pi \right] \), find the value of \( x_0 \).

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