
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 that the distance from moving point $P$ to point $F(\sqrt{5},0)$ is $\sqrt{5}$ times the distance to the line $x=\frac{\sqrt{5}}{5}$, let the trajectory of moving point $P$ be denoted as curve $\Gamma$.  
$(1)$ Find the equation of $\Gamma$;  
$(2)$ Can a line $l$ be drawn passing through point $A(1,1)$ such that $l$ intersects $\Gamma$ at points $B$ and $C$, and $A$ is the midpoint of segment $BC$? If it exists, find the equation of line $l$; if not, explain the reason.

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