
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
In the Cartesian coordinate system $xOy$, the perimeter of $\triangle ABC$ is $12$, and the midpoints of sides $AB$ and $AC$ are $F_{1}(-1,0)$ and $F_{2}(1,0)$, respectively. Point $M$ is the midpoint of side $BC$.
$(1)$ Find the equation of the trajectory of point $M$;
$(2)$ Let the trajectory of point $M$ be curve $T$, the line $MF_{1}$ intersects curve $T$ at another point $N$, and the midpoint of segment $MF_{2}$ is $E$. Let $S=S_{\triangle NF_{1}O}+S_{\triangle MF_{1}E}$, find the maximum value of $S$.

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