
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 the parabola $C: y^2 = 4x$, point $M(m,0)$ is on the positive half of the $x$-axis, and line $l$ passing through point $M$ intersects the parabola $C$ at points $A$ and $B$, with $O$ being the origin.

$(I)$ If $m=1$, and the slope of line $l$ is $1$, find the equation of the circle with diameter $AB$;

$(II)$ Is there a fixed point $M$, such that no matter how line $l: x=ky+m$ rotates around point $M$, $\frac{1}{{|AM|}^2}+ \frac{1}{{|BM|}^2}$ always equals a constant value?

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