
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
In a number system with base \( r \) ( \( r \leq 100 \) ), a natural number \( x \) is a two-digit number with identical digits. It turns out that the \( r \)-ary notation of \( x^2 \) is a four-digit number where the end digits are the same and the middle digits are zero. For which \( r \) is this possible?

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