
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 $F_1(-2, 0)$, $F_2(2, 0)$, and a point $P$ satisfies $|PF_1| - |PF_2| = 2$, denote the trajectory of point $P$ as $\Gamma$. A line $l$ with slope $k$ passes through point $F_2$ and intersects the trajectory $\Gamma$ at points $A$ and $B$.
1. Find the equation of the trajectory $\Gamma$;
2. Determine the range of values for the slope $k$;
3. Is there a fixed point $M$ on the x-axis such that $MA \perp MB$ holds regardless of how the line $l$ rotates around point $F_2$? If it exists, find the fixed point $M$; if not, explain why.

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