
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 the Cartesian coordinate system $xOy$, points $A(-2,0)$, $B(6,0)$, and point $C$ is on the line $x=6$. A line through the midpoint $D$ of $AB$ and perpendicular to $OC$ intersects $AC$ at point $P$. The trajectory of point $P$ is curve $\Gamma$.
$(1)$ Find the equation of the trajectory of $\Gamma$;
$(2)$ A line $l$ passing through point $Q(2, \sqrt{3})$ intersects $\Gamma$ at points $E$ and $F$. The line $x=x_0$ intersects lines $DE$ and $DF$ at points $S$ and $T$ respectively. Determine whether the midpoint $M$ of segment $ST$ is on a fixed line. If yes, find the equation of this line; if not, explain why.

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