
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 $F$ is the focus of the parabola $x^2=y$, and points $M$, $N$ are on the parabola and located on opposite sides of the $x$-axis. $\overrightarrow{OM} \cdot \overrightarrow{ON}=2$, where $O$ is the origin, determine the minimum value of the sum of the areas of $\triangle MON$ and $\triangle MFO$.

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