
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 begins by writing the square of the integer just three more than the smallest \(k\)-digit number, i.e., \((10^{k-1} + 3)^2\). Bernardo writes every third perfect square thereafter. Silvia then erases the last \(k-1\) digits of each number Bernardo writes. This process continues until the last two numbers that remain differ by at least 2. Define \(f(k)\) as the smallest integer that does not appear on the board. What is \(f(3)\)?
A) 100
B) 101
C) 102
D) 103
E) 104

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