
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 $k$ be a positive integer. Bernardo and Silvia play a game where Bernardo starts by writing the smallest perfect square with $k+2$ digits on a blackboard. Each turn, Bernardo writes the next perfect square, and Silvia erases the last $k+1$ digits of the current number on the board after multiplying it by $3$. They stop when the difference between two remaining numbers on the board after Silvia's action is at least $3$. Define $f(k)$ as the smallest positive integer not written on the board during this game. Calculate the value of $f(1)$.

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