
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
The directrix of the parabola $C: y^2 = 2x$ is ___________. A line $l$ passes through point $P(4,1)$ and intersects the parabola $C$ at points $A$ and $B$, with $P$ being the midpoint of segment $AB$. Let $F$ be the focus of the parabola. Find the value of $|\vec{AF}| + |\vec{BF}|$.

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