
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  $m$  be a positive integer with  $m \leq 2024$ . Ana and Banana play a game alternately on a  $1\times2024$  board, with squares initially painted white. Ana starts the game. Each move by Ana consists of choosing any  $k \leq m$  white squares on the board and painting them all green. Each Banana play consists of choosing any sequence of consecutive green squares and painting them all white. What is the smallest value of  $m$  for which Ana can guarantee that, after one of her moves, the entire board will be painted green?

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