
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 \(x\) and \(y\) be two distinct positive real numbers, and \(n\) a positive integer. We define sequences \((P_n), (G_n)\) as follows. First, \(P_1\) is \( \sqrt[n]{\frac{x^n + y^n}{2}} \) and \( G_1 \) is \( \sqrt{x \cdot y} \). Then for \( n \ge 2 \), \( P_n \) and \( G_n \) are:
\[ P_{n+1} = \sqrt[n]{\frac{(\sin P_n)^n + (\sin G_n)^n}{2}} \text{ and } G_{n+1} = \sqrt{\sin P_n \cdot \sin G_n} \] respectively.

Determine if the sequences \( (P_n) \) and \( (G_n) \) converge and identify their limits if they exist.

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