
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

Consider a graph where the vertices correspond to all possible three-element subsets of the set $\{1, 2, 3, \ldots, 2^k\}$, and edges are drawn between vertices that correspond to subsets intersecting in exactly one element. Find the minimum number of colors needed to color the vertices of the graph such that any two vertices connected by an edge are of different colors.

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