
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 points $A$ and $B$ on the parabola $y^{2}=4x$ that are distinct from the vertex $O$, the product of the slopes of lines $OA$ and $OB$ is a constant (-4). $F$ represents the focus of the parabola, and the areas of triangles $AOF$ and $BOF$ are $S_{1}$ and $S_{2}$, respectively. Determine the minimum value of $S_{1}^{2}+S_{2}^{2}$.

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