
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 with the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ whose eccentricity is $\frac{\sqrt{2}}{2}$, and one endpoint of its minor axis is $M(0,1)$. A line $l$ passing through the left vertex $A$ of the ellipse intersects the ellipse at another point $B$.
(1) Find the equation of the ellipse;
(2) If line $l$ intersects line $x=a$ at point $P$, find the value of $\overrightarrow{OB}\cdot\overrightarrow{PO}$;
(3) If $|AB| = \frac{4}{3}$, find the acute angle that line $l$ makes with the positive direction of the $x$-axis.

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