
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 $C: y^{2}=2px(p > 0)$, the distance from point $P(t,2)$ in the first quadrant to the focus $F$ is $\frac{5}{2}$, and the projection of vector $\overrightarrow{FP}$ on vector $\overrightarrow{OF}$ is positive ($O$ is the origin).
- (I) If $M(-\frac{1}{2},0)$, and a line $l_{1}$ through points $M$ and $P$ intersects the parabola at another point $Q$, find the value of $\frac{|QF|}{|PF|}$.
- (II) If a line $l_{2}$ intersects the parabola $C$ at points $A$ and $B$, and intersects the circle $M: (x-a)^{2}+y^{2}=1$ at points $D$ and $E$, with $OA \perp OB$, is there a real number $a$ such that the length of $|DE|$ is constant? If so, find the value of $a$; if not, explain why.

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