
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 a parabola $x^{2}=2py$ $(p>0)$ and a circle $x^{2}+y^{2}=r^{2}$ $(r>0)$, the chord they share passes through the focus $F$ of the parabola, and the chord length is $4$.
$(1)$ Find the equations of the parabola and the circle.
$(2)$ A line passing through the point $F$ intersects the parabola at points $A$ and $B$, the tangent to the parabola at point $A$ intersects the x-axis at point $M$. Find the minimum value of the area of triangle $ABM$.

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