
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 $xOy$, the equation of circle $C$ is given by $x^2 + y^2 - 8x + 11 = 0$, and the equation of line $l$ is $(2m+1)x + (m+1)y - 7m - 4 = 0$.

$(1)$ When $m=1$, find the length of the chord intercepted by circle $C$ on line $l$;

$(2)$ When the length of the chord intercepted by circle $C$ on line $l$ is the shortest, find the equation of line $l$;

$(3)$ Under the premise of $(2)$, if $P$ is a moving point on line $l$, and there exist two distinct points on circle $C$ such that the distance from each to point $P$ is $\sqrt{5}$, find the range of the $x$-coordinate of point $P$.

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