
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

Sergio asked Ivan to think of a positive integer. Then, he asked Ivan to calculate the sum of its digits and, finally, square the result. Without revealing the initial number, Ivan stated that the final result was $x$. Demonstrate the following conclusions to Sergio:

a) If Ivan had thought of a number with 3 or fewer digits, then $x$ would be less than 730.

b) If Ivan had thought of a number with 4 digits, then $x$ would be less than the number Ivan thought of.

c) If Ivan had thought of a number with 5 or more digits, then $x$ would be less than the number Ivan thought of.

Later, Sergio did the following: He considered the number $x$ that Ivan mentioned, calculated the sum of its digits, and squared the result. When Sergio told Ivan the number he obtained, Ivan was surprised to learn that it was the number he had originally thought of.

d) Determine all possible values for the number Ivan initially thought of.

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