
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 $C$ with its vertex at the origin and focus on the $y$-axis. A point $Q(a, 2)$ on the parabola $C$ is 3 units away from the focus. The endpoints of line segment $AB$, denoted by $A(x_1, y_1)$ and $B(x_2, y_2)$, are also on the parabola $C$.

(1) Find the equation of parabola $C$.
(2) If there exists a point $M(0, m)$ on the $y$-axis ($m > 0$) such that line segment $AB$ passes through point $M$ and the circle with $AB$ as its diameter goes through the origin, find the value of $m$.
(3) There is a point $D(x_3, y_3)$ on parabola $C$ that satisfies $x_3 < x_1 < x_2$. If $\triangle ABD$ is an isosceles right triangle with a right angle at $A$, find the minimum area of $\triangle ABD$.

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