
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 with the equation $x^{2}=2py (p > 0)$ and focus $F$, the point $P$ intersects with the line $x=4$ and the $x$-axis, and the point $Q$ intersects with the line $x=4$ and the parabola, such that $|QF|= \frac{5}{4}|PQ|$.

(1) Find the equation of the parabola;

(2) A line $l$ passes through $F$ and intersects with the parabola at points $A$ and $D$, and with the circle ${x}^{2}+(y-1{)}^{2}=1$ at points $B$ and $C$ (with $A$ and $B$ being adjacent). If tangent lines are drawn from $A$ and $D$ to the parabola, intersecting at point $M$, find the minimum value for the product of the areas of triangles $ABM$ and $CDM$.

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