
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
In triangle \( \triangle ABC \), it is known that \( AB = 2 \), \( AC = 1 \), and \( \cos 2A + 2 \sin^2 \frac{B+C}{2} = 1 \).

(1) Find the measure of angle \( A \) and the length of side \( BC \);

(2) If point \( P \) moves inside \( \triangle ABC \) (including the boundary) and the sum of the distances from point \( P \) to the three sides is \( d \), let the distance from point \( P \) to sides \( BC \) and \( CA \) be \( x \) and \( y \) respectively. Express \( d \) in terms of \( x \) and \( y \), and determine the range of values \( d \) can take.

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