
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 $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C: \frac{{x}^{2}}{{a}^{2}} + y^{2} = 1 (a > 1)$, and the point $F_{2}$ is symmetric to the point $Q$ about the line $y=x$ and lies on the ellipse. Find the length of the major axis. If $P$ is a point on the ellipse and $|PF_{1}| \cdot |PF_{2}| = \frac{4}{3}$, then find the area of $\triangle F_{1}PF_{2}$.

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