
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 $y^{2}=2px$ and the line $ax+y-4=0$ intersect at points $A$ and $B$, where the coordinates of point $A$ are $(1,2)$. If the focus of the parabola is $F$, calculate $|FA|+|FB|$.

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