
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 to make the resulting 12-digit number a perfect square?

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

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

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