
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
Given Bernardo and Silvia play a game starting with an integer between $0$ and $499$ inclusive, where Bernardo doubles the number and passes it to Silvia, Silvia adds $75$ to the number she receives and passes it back to Bernardo, and the last number Bernardo produces less than $500$, determine the smallest initial number $N$ such that Bernardo wins the game, and find the sum of the digits of $N$.

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