
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 that the focus of the parabola $C:y^{2}=4x$ is $F$, the line $l$ passing through the point $P\left(5,0\right)$ intersects $C$ at points $A$ and $B$. Let $O$ be the origin. Denote the areas of triangles $ABO$ and $AFO$ as $S_{1}$ and $S_{2}$, respectively. Compute the minimum value of $S_{1}+3S_{2}$.

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