
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. Two people, A and B, take turns either writing or erasing a number on the blackboard. The rules are as follows: A starts by writing the smallest \( k+1 \)-digit perfect square on the blackboard. Each time A writes a \( k+1 \)-digit perfect square, B erases the last \( k \) digits of this number. This process continues until there are two consecutive numbers on the blackboard that differ by at least 2. Let \( f(k) \) be the smallest positive integer that is not written on the blackboard. Calculate the sum of the digits of \(\sum_{k=1}^{1008} f(2k)\).

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