
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 $y^{2}=2px\left(x \gt 0\right)$, point $P\left(2,1\right)$ lies inside the parabola. A line $l:y=2x+m$ that does not pass through point $P$ intersects the parabola at points $A$ and $B$. Lines $AP$ and $BP$ intersect the parabola at points $C$ and $D$, respectively. If for any line $l$, there exists a $\lambda$ such that $\overrightarrow{AP}=λ\overrightarrow{PC}$ and $\overrightarrow{BP}=λ\overrightarrow{PD}(\lambda\ \ \gt 0,\lambda \neq 1)$ hold, then $p=\_\_\_\_\_\_$.

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