
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
Let  $G$  be a simple, undirected, connected graph with  $100$  vertices and  $2013$  edges. It is given that there exist two vertices  $A$  and  $B$  such that it is not possible to reach  $A$  from  $B$  using one or two edges. We color all edges using  $n$  colors, such that for all pairs of vertices, there exists a way connecting them with a single color. Find the maximum value of  $n$ .

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