
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 natural number \( x \) in a numeral system with base \( r \) (where \( r > 3 \)) has the form \(\overline{p p q q}\), where \( q = 2p \). It turns out that the base-\( r \) representation of the number \( x^{2} \) is a seven-digit palindrome with three identical middle digits. (A palindrome is a number that reads the same from left to right and right to left). For which values of \( r \) is this possible?

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