
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
If $M(\frac{p}{2}$,$p)$ is a point on the parabola $y^{2}=2px\left(p \gt 0\right)$, and the distance from $M$ to the point $\left(1,0\right)$ is 1 greater than the distance to the $y$-axis, find:<br/>
$(1)$ The equation of the parabola.<br/>
$(2)$ Let line $l$ intersect the parabola at points $A$ and $B$. The circle with diameter $AB$ passes through point $M$. Find the maximum distance from point $\left(1,0\right)$ to line $l$.

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