
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 two circles $F_1: (x+1)^2+y^2=t^2$ and $F_2: (x-1)^2+y^2=(2\sqrt{2}-t)^2$, where $0<t<2\sqrt{2}$, when the two circles have common points, the curve formed by all possible common points is denoted as $C$.
1. Find the equation of curve $C$.
2. Given vector $\overrightarrow{a}=(1, \sqrt{3})$, and $M$, $N$, $P$ are three distinct points on curve $C$, with $\overrightarrow{F_{2}M}=\lambda\overrightarrow{a}$ and $\overrightarrow{F_{2}N}=\mu\overrightarrow{a}$, find the maximum area of $\triangle PMN$.

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