
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 points $F_{1}(-1,0)$, $F_{2}(1,0)$, and a moving point $P$ satisfying the relationship $|PF_{1}|+|PF_{2}|=4$.  
$(1)$ Find the equation of the trajectory $C$ of the moving point $P$;  
$(2)l$ is a line passing through point $F_{1}(-1,0)$ with a slope of $2$, $M$ is a moving point on trajectory $C$ (not on line $l$), point $A$ lies on line $l$, and $MA\bot l$. Find the maximum value of $|F_{1}A|$ and the coordinates of point $M$ at that moment.

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