
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 $C$: $y^{2}=8x$ with a focus at $F$, and a point $P$ on the directrix of $C$. Let $Q$ (in the first quadrant) be an intersection point of line $PF$ and $C$. If $\overrightarrow{PQ} = \sqrt{2}\overrightarrow{QF}$, find the length of $QF$.

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