
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
Given an ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ with $a > b > 0$, draw a perpendicular line from the right focus F<sub>2</sub> to the x-axis that intersects ellipse C at point P. If $\sin\angle PF_1F_2 = \frac{1}{3}$, determine the relationship between $a$ and $b$.

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