
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 $C: \frac{x^{2}}{t} + \frac{y^{2}}{t-1} = 1 (t > 1)$ with two foci $F_1$ and $F_2$. $P$ is one of the ellipse's minor axis vertices, and $\angle{PF_1F_2} = 60^\circ$.

1. Find the standard equation of the ellipse $C$.
2. If a line $l$ passing through the left focus of ellipse $C$ intersects the ellipse at points $A$ and $B$, and $Q$ is the right vertex of ellipse $C$, find the maximum area of $\triangle{ABQ}$.

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