
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 where they alternately take marbles from the table. Anna makes the first move. If at the beginning of a turn there are \( n \geq 1 \) marbles on the table, the player whose turn it is takes \( k \) marbles, where \( k \geq 1 \) is either an even number with \( k \leq \frac{n}{2} \) or an odd number with \( \frac{n}{2} \leq k \leq n \). A player wins the game if they take the last marble from the table.

Determine the smallest number \( N \geq 100000 \) such that Berta can force a win if there are initially exactly \( N \) marbles on the table.

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