
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]


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A tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other.  Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges  $\overrightarrow{uv}$  and  $\overrightarrow{vw}$ , those two edges are in different colors.  Note that it is permissible for  $\overrightarrow{uv}$  and  $\overrightarrow{uw}$  to be the same color.  The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring.  For each  $n$ , determine the minimum directed-edge-chromatic-number over all tournaments on  $n$  vertices.

*Proposed by Po-Shen Loh*

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