
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 that $O$ is the coordinates origin, and $C$ is the parabola: $y^{2}=nx (n > 0)$ in the first quadrant, the point $P(2,t)$ on the parabola is a distance of $\frac{5}{2}$ from the focus. The tangent line of the curve $C$ at point $P$ intersects the $x$-axis at point $Q$, and line $l_{1}$ passes through point $Q$ and is perpendicular to the $x$-axis.

(1) Find the coordinates of point $Q$;

(2) Let the moving line $l_{2}$: $x=my+b$ intersect 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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