
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 O as the origin, the parabola $C: y^2 = nx$ ($n > 0$) has a point $P(2, t)$ in the first quadrant whose distance to the focus is $\frac{5}{2}$. The tangent to the curve $C$ at point $P$ intersects the x-axis at point $Q$. Let line $l_1$ pass through point $Q$ and be perpendicular to the x-axis.
1. Find the length of the line segment $OQ$.
2. Let a moving line $l_2: x = my + b$ that does not pass through points $P$ and $Q$ intersect the curve $C$ at points $A$ and $B$, and intersect $l_1$ at point $E$. If the slopes of lines $PA$, $PE$, and $PB$ form an arithmetic sequence, does $l_2$ pass through a fixed point? Please explain your reasoning.

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