
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
In the Cartesian coordinate system $xOy$, let the moving point P be equidistant from a fixed point $F(1,0)$ and from a fixed line $l: x=-1$. Let the trajectory of P be denoted by $\Gamma$. Also, the line $AB$ has a direction vector $\vec{d}=(1,2)$ and passes through the point $(1,0)$. The line $AB$ intersects $\Gamma$ at points A and B. Find the length of $|AB|$.

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