
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 $C: y^2 = 2px$ ($p > 1$) with focus $F$, the line $y = m$ intersects the y-axis at point $P$, and intersects $C$ at point $Q(x_0, y_0)$, and $\frac {|QF|}{|PQ|} = p$.  
(1) When $x_0 + p$ reaches its minimum value, find the value of $p$;  
(2) When $x_0 = 1$, if line $l$ intersects the parabola $C$ at points $A$ and $B$, and intersects the circle $M: (x-n)^2 + y^2 = 1$ at points $D$ and $E$, with $O$ being the origin, and $OA \perp OB$, is there a real number $n$ such that the length of $|DE|$ is constant? If so, find the value of $n$; if not, explain why.

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