
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 a parabola $C$, let a line $l$ pass through the focus $F$ of $C$, and the angle between $l$ and the axis of symmetry of $C$ is $\frac{\pi}{4}$. If the length of the chord intercepted by $l$ on $C$ is $4$, find the distance from the focus to the vertex of the parabola $C$.

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