
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passing through two points $P(1, \frac{\sqrt{2}}{2})$ and $Q(-\sqrt{2}, 0)$.  
$(Ⅰ)$ Find the standard equation of the ellipse $C$;  
$(Ⅱ)$ A line $l$ passing through the right focus $F$ of the ellipse intersects the ellipse $C$ at points $A$ and $B$, and intersects a circle with diameter $FP$ at another point $E$ (different from point $F$). Find the maximum value of $|AB| \cdot |FE|$.

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