
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 $E$ with foci $F_{1}(-1,0)$, $F_{2}(1,0)$, and eccentricity $e= \frac { \sqrt {2}}{2}$.  
$(1)$ Find the equation of the ellipse $E$;  
$(2)$ Let line $l$: $y=x+m(m\neq 0)$ intersect ellipse $E$ at points $A$ and $B$, and the perpendicular bisector of segment $AB$ intersects the $x$-axis at point $T$. When $m$ varies, find the maximum value of the area of $\triangle TAB$.

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