
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 the parabola $y^2 = 2x$, a line $l$ passes through point A($-2$, $4$) and intersects the parabola at points B and C. Let O be the origin, and P($m$, $0$) with $m > 0$, such that $\tan \angle PAO = \frac{2}{21}$.
(1) Find the value of $m$.
(2) If the areas of triangles $\triangle PAB$, $\triangle PBC$, and $\triangle PAC$ form a geometric sequence, find the equation of line $l$.

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