
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 \( A B C \) be a right triangle with \(\angle BAC = 90^{\circ}\) and \(I\) the intersection point of its angle bisectors. A line through \(I\) intersects the sides \(AB\) and \(AC\) at \(P\) and \(Q\) respectively. The distance from \(I\) to the side \(BC\) is \(1 \, \text{cm}\).

a) Find the value of \( PM \cdot NQ \).

b) Determine the minimum possible value for the area of triangle \(APQ\).

Hint: If \(x\) and \(y\) are two non-negative real numbers, then \(x + y \geq 2 \sqrt{xy}\).

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