
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 the ellipse $C$: $x^{2}+3y^{2}=6$ with its right focus $F$.
(I) Find the coordinates of point $F$ and the eccentricity of ellipse $C$;
(II) Line $l$: $y=kx+m(k\neq 0)$ passes through point $F$ and intersects ellipse $C$ at points $P$ and $Q$. If the point $P'$ symmetric to $P$ about the $x$-axis, determine whether line $P'Q$ passes through a fixed point on the $x$-axis. If it does, find the coordinates of this point; if not, explain the reason.

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