
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 $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) whose left focus $F_1$ coincides with the focus of the parabola $y^2 = -4x$, and the eccentricity of ellipse $E$ is $\frac{\sqrt{2}}{2}$. A line $l$ with a non-zero slope passes through point $M(m, 0)$ ($m > \frac{3}{4}$) and intersects ellipse $E$ at points $A$ and $B$. Point $P(\frac{5}{4}, 0)$, and $\overrightarrow{PA} \cdot \overrightarrow{PB}$ is a constant.  
(Ⅰ) Find the equation of ellipse $E$;  
(Ⅱ) Find the maximum value of the area of $\triangle OAB$.

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