
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 \( a^{\frac{2}{3}} + b^{\frac{2}{3}} = 17 \frac{1}{2} \), \( x = a + 3a^{\frac{1}{3}} b^{\frac{2}{3}} \) and \( y = b + 3a^{\frac{2}{3}} b^{\frac{1}{3}} \). If \( P = (x+y)^{\frac{2}{3}} + (x-y)^{\frac{2}{3}} \), find the value of \( P \).

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