
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 $F_{2}: (x-1)^{2}+y^{2}=(2 \sqrt {2}-t)^{2}$, with $0 < t < 2 \sqrt {2}$. 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})$, $M$, $N$, and $P$ are three different points on curve $C$, and $ \overrightarrow{F_{2}M}=\lambda \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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