
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
Let \( A \) be the center of a circle with radius \( 2 \cos \theta \left(\frac{\pi}{4} < \theta < \frac{\pi}{2}\right) \) and there is a point \( B \) outside the circle such that \( |AB| = 2 \sin \theta \). Suppose the center of the circle, which passes through point \( B \) and is tangent to the circle centered at \( A \) at point \( T \), is denoted as \( M \).

1. When \(\theta\) takes a specific value, what is the curve described by the trajectory \( P \) of point \( M \)?
2. Let \( M \) be a moving point on the trajectory \( P \), and let \( N \) be a moving point on the circle centered at \( A \). Denote the minimum value of \( |MN| \) as \( f(\theta) \) (proof not required). Find the range of values taken by \( f(\theta) \).
3. If the range of \(\theta\) is modified to \(\left(0 < \theta < \frac{\pi}{4}\right)\) and the position of point \( B \) is changed to inside the circle centered at \( A \) while other conditions remain unchanged, denote the trajectory of point \( M \) as \( P \). Propose a meaningful question that possesses a similar structure to (2) (solution not required).

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