
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 line $y=2x$ and the parabola $\Gamma$: $y^{2}=2px$ intersect at points $O$ and $E$, and $|OE|= \sqrt {5}$.  
$(1)$ Find the equation of the parabola $\Gamma$;  
$(2)$ A line passing through point $Q(2,0)$ intersects the parabola $\Gamma$ at points $A$ and $B$. Point $P$ is on the line $x=-2$, and lines $PA$, $PB$ intersect the $x$-axis at points $M$ and $N$ respectively. Is the product of the $x$-coordinates of points $M$ and $N$, $x_{M} \cdot x_{N}$, a constant? If it is, find the constant; otherwise, explain why.

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