
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 the plane rectangular coordinate system \( xOy \), the ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) with \( a > b > 0 \) has its left and right foci at \( F_{1} \) and \( F_{2} \) respectively. The chords \( ST \) and \( UV \) of the ellipse \( C \) are parallel to the \( x \)-axis and \( y \)-axis, respectively, and intersect at point \( P \). Given that the lengths of segments \( PU, PS, PV, \) and \( PT \) are \( 1, 2, 3, \) and \( 6 \) respectively, find the area of triangle \( \triangle PF_{1}F_{2} \).

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