
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
In $\triangle ABC$, $A(-1,0)$, $B(1,0)$, if the centroid $G$ and the orthocenter $H$ of $\triangle ABC$ satisfy $GH$ is parallel to the $x$-axis ($G$ and $H$ do not coincide),
(I) Find the equation of the trajectory $\Gamma$ of the moving point $C$;
(II) Given that $O$ is the coordinate origin, if line $AC$ is tangent to the circle with center $O$ and radius $|OH|$, find the equation of line $AC$ at this time.

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