
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
Find the center of gravity of a plate bounded by the lines \( x=0 \), \( y=0 \), \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \), \( y \geqslant 0 \), \( x \geqslant 0 \), given that the surface density \( P = x y \).

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