
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
Each positive integer  $a$  undergoes the following procedure in order to obtain the number  $d = d\left(a\right)$ :

(i) move the last digit of  $a$  to the first position to obtain the numb er  $b$ ;
(ii) square  $b$  to obtain the number  $c$ ;
(iii) move the first digit of  $c$  to the end to obtain the number  $d$ .

(All the numbers in the problem are considered to be represented in base  $10$ .)  For example, for  $a=2003$ , we get  $b=3200$ ,  $c=10240000$ , and  $d = 02400001 = 2400001 = d(2003)$ .)

Find all numbers  $a$  for which  $d\left( a\right) =a^2$ .

*Proposed by Zoran Sunic, USA*

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