
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
Let $\mathcal{X}$ be the set of 3-connected graphs and $G$ be a graph with a true separation $\left\{V_{1}, V_{2}\right\}$ such that $\kappa(G) \leq 2$. If $G$ does not contain any topological minor from $\mathcal{X}$ and is edge-maximal, then $G_{1} := G\left[V_{1}\right]$ and $G_{2} := G\left[V_{2}\right]$ are also edge-maximal, and $G_{1} \cap G_{2} = K^{2}$.

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