
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
There are 10,000 vertices in a graph, with at least one edge coming out of each vertex. Call a set  $S{}$  of vertices *delightful* if no two of its vertices are connected by an edge, but any vertex not from  $S{}$  is connected to at least one vertex from  $S{}$ . For which smallest  $m$  is there necessarily a delightful set of at most  $m$  vertices?

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