
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
A tournament graph is a directed graph where every pair of distinct vertices has a single directed edge connecting them. Define a proper directed edge coloring as an assignment of colors to each directed edge such that every two directed edges $\overrightarrow{uv}$ and $\overrightarrow{vw}$ (where the endpoint of the first edge and the start point of the second edge are the same vertex $v$) have different colors. However, it is allowed for pairs of edges $\overrightarrow{vu}$ and $\overrightarrow{vw}$ to have the same color; similarly, for pairs $\overrightarrow{uv}$ and $\overrightarrow{wv}$ to have the same color. The edge coloring number of a tournament graph is the minimum number of colors needed to properly color the directed edges. For each positive integer $n$ and for all tournament graphs with $n$ vertices, find the minimum edge coloring number $f(n)$.

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