
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

The number 76 has an interesting property: the last two digits of $76^2 = 5776$ are 76 again.

a) Are there any other two-digit numbers with this property?

b) Find all three-digit numbers $A$ such that the last three digits of $A^2$ form the number $A$.

c) Does there exist an infinite sequence of digits $a_{1}, a_{2}, a_{3}, \ldots$, such that for any natural number $n$, the square of the number formed by $\overline{a_{n}} \overline{a_{n-1}} \bar{\ldots} \bar{\ldots} \overline{\overline{2}} \overline{a_{1}^{-}}$ ends in these same $n$ digits? The obvious answer $a_{1} = 1$ and $a_{2} = a_{3} = \ldots = 0$ is excluded.

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