
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
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are  $n\geq 1$  marbles on the table, then the player whose turn it is removes  $k$  marbles, where  $k\geq 1$  either is an even number with  $k\leq \frac{n}{2}$  or an odd number with  $\frac{n}{2}\leq k\leq n$ . A player win the game if she removes the last marble from the table.
Determine the smallest number  $N\geq 100000$  such that Berta can enforce a victory if there are exactly  $N$  marbles on the tale in the beginning.

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