
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 $P$ lies on the ellipse $C$ with foci $F_1$ and $F_2$, and $|PF_1|$, $|F_1F_2|$, $|PF_2|$ form an arithmetic sequence.  
(I) Find the equation of the ellipse $C$;  
(II) Suppose a line $m$ passing through $F_2$ intersects 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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