
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, given that the directrix of the parabola $y^{2}=2px$ is $x=-\frac{1}{2}$.  
$(1)$ Find the value of $p$;  
$(2)$ The line $l:y=x+t\left(t\neq 0\right)$ intersects the parabola at points $A$ and $B$, with the origin $O$ as the coordinate origin, and $OA\bot OB$. Find the length of the line segment $AB$.

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