
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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) with eccentricity $\frac{\sqrt{2}}{2}$, and it passes through the point (0, 1).  
(I) Find the equation of the ellipse;  
(II) If a line $l$ passing through the left vertex $A$ of the ellipse intersects the ellipse at another point $B$ and intersects the line $x = a$ at point $P$, find the value of $\overrightarrow{OB} \cdot \overrightarrow{OP}$.

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