
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
a) Is it possible to append 6 digits to any six-digit number starting with the digit 5 so that the resulting 12-digit number is a perfect square?

b) Is the same possible for a number starting with the digit 1?

c) Find the smallest $k = k(n)$ for each $n$ such that it is possible to append $k$ digits to any $n$-digit number so that the resulting $(n+k)$-digit number is a perfect square.

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