
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
Initially, all edges of the  $K_{2024}$  are painted with  $13$  different colors. If there exist  $k$  colors such that the subgraph constructed by the edges which are colored with these  $k$  colors is connected no matter how the initial coloring was, find the minimum value of  $k$ .

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