
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) Can any six-digit number starting with the digit 5 be appended with 6 more digits so that the resulting 12-digit number is a perfect square?

b) The same question for a number starting with the digit 1.

c) Find for each \( n \) the smallest \( k = k(n) \) such that any \( n \)-digit number can be appended with \( k \) more digits 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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