
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 eccentricity $\frac{\sqrt{2}}{2}$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ that passes through point $A(1, \frac{\sqrt{2}}{2})$.

$(1)$ Find the equation of ellipse $E$;  
$(2)$ If a line $l$: $y= \frac{\sqrt{2}}{2}x+m$ that does not pass through point $A$ intersects ellipse $E$ at points $B$ and $C$, find the maximum value of the area of $\triangle ABC$.

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