
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
Determine the smallest initial number, $N$, that results in a win for Bernardo in the game, where a number between $0$ and $999$ inclusive is selected, doubled by Bernardo, added by $100$ by Silvia, and repeated until a number less than $1000$ is produced, and calculate the sum of the digits of $N$.

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