
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
A parabola has its vertex at the origin of the coordinate system, and its focus $F$ is located on the positive half of the $y$-axis. A line passing through point $F$ intersects the parabola at points $A$ and $B$. The length of the line segment $AB$ is $8$, and the distance from the midpoint of $AB$ to the $x$-axis is $3$.

(I) Find the standard equation of the parabola;
(II) Suppose a line $m$ has a $y$-intercept of $6$ and intersects the parabola at points $P$ and $Q$. Connect $QF$ and extend it to intersect the parabola's directrix at point $R$. When line $PR$ is tangent to the parabola, find the equation of line $m$.

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