
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 $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$. It passes through the point $P(1, \frac{ \sqrt{2}}{2} )$, and the eccentricity of $E$ is $\frac{ \sqrt{2}}{2}$.

(I) Find the equation of $E$;

(II) Two mutually perpendicular lines are drawn through the vertex $A(0,b)$ of the ellipse $E$ and intersect the ellipse at points $B$ and $C$, respectively. If the angle bisector of $\angle BAC$ has the equation $y=-3x+1$, find the area of $\triangle ABC$.

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