
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 two points $F\_1(-1,0)$ and $F\_2(1,0)$, point $P$ is on the ellipse $C$ with foci $F\_1$ and $F\_2$. The distances $|PF\_1|$, $|F\_1F\_2|$, and $|PF\_2|$ form an arithmetic sequence.

(I) Find the equation of the ellipse $C$.
(II) Let line $m$ pass through $F\_2$ and intersect curve $C$ at points $P$ and $Q$. If $|PQ|^2 = |F\_1P|^2 + |F\_1Q|^2$, find the equation of line $m$.

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