
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
Adithya and Bill are playing a game on a connected graph with \(n > 2\) vertices and \(m\) edges. First, Adithya labels two of the vertices \(A\) and \(B\), such that \(A\) and \(B\) are distinct and non-adjacent, and announces his choice to Bill. Then Adithya starts on vertex \(A\) and Bill starts on \(B\).

The game proceeds in a series of rounds in which both players move simultaneously. In each round, Bill must move to an adjacent vertex, while Adithya may either move to an adjacent vertex or stay at his current vertex. Adithya loses if he is ever on the same vertex as Bill, and wins if he reaches \(B\) alone. Adithya cannot see where Bill is, but Bill can see where Adithya is.

Given that Adithya has a winning strategy, what is the maximum possible value of \(m\), in terms of \(n\)?

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