
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 $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and an eccentricity of $\frac{\sqrt{2}}{2}$. Let $F\_1$ and $F\_2$ be the foci of the ellipse, and let $P$ be a point on the ellipse.

(I) If $\angle F\_1PF\_2 = 90^{\circ}$ and the area of $\triangle PF\_1F\_2$ is equal to $1$, find the equation of the ellipse.

(II) If the line $PF\_1$ intersects the ellipse at another point $Q$, and lines perpendicular to $PQ$ are drawn through points $P$ and $Q$ intersecting the $x$-axis at points $M$ and $N$, find the slope of the line $PQ$ when $|MN|$ is at its minimum value.

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