
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
Professor Rackbrain recently asked his young friends to find all five-digit perfect squares for which the sum of the numbers formed by the first two digits and the last two digits equals a perfect cube. For example, if we take the square of the number 141, which is equal to 19881, and add 81 to 19, we get 100 – a number that unfortunately is not a perfect cube.

How many such solutions exist?

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