
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
 $\textbf{Problem 4.}$  The number of perfect inhabitants of a city was a perfect square, in other words, a whole number squared. with  $100$  people plus the new number of inhabitants turned out to be a perfect square plus one. Now, with another increase of  $100$  people, the number of inhabitants is again a perfect square. What was the number of inhabitants original city?

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