
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, symmetric about the $x$-axis, and passing through point $P(1, 2)$.
(I) Find the standard equation of the parabola $C$ and the equation of its directrix;
(II) Given a line $l$ that does not pass through point $P$ and has a slope of $1$, and intersects the parabola $C$ at points $A$ and $B$. If the circle with $AB$ as its diameter passes through point $P$, find the equation of line $l$.

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