
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 $C: y^2 = 2px (p > 0)$ with a focus at $F$, and a circle centered at a moving point $M$ on the parabola passing through $F$, find:
(Ⅰ) The value of $p$ if the minimum area of the circle $M$ is $\pi$;
(Ⅱ) When the $x$-coordinate of point $M$ is 1 and it lies in the first quadrant, construct two chords $MA$, $MB$ of the parabola through $M$, such that $\angle AMF = \angle BMF$. If the line $AB$ is tangent to the circle $M$, find the equation of the line $AB$.

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