
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 $C: x^2 = 2py (p > 0)$, its focus is at $F(0,1)$.
$(1)$ Find the value of $p$;
$(2)$ A line $l$ passing through point $F$ intersects the parabola $C$ at points $A$ and $B$. A circle with diameter $AB$ intersects the x-axis at points $M$ and $N$, with midpoint $Q$. Find the minimum value of angle $\angle QMN$ and the equation of line $l$ at this condition.

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