Doraemon's Challenge

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Difficulty: 8

This problem is tagged with miscellaneous, coolmath.

Nobita, Doraemon, and their friends have discovered a mysterious time-traveling gadget. However, there's a catch – the gadget can only be activated if a specific mathematical condition is met. Doraemon, being the math whiz, has set up the following challenge for them:
The gadget will only activate if the sum of the cubes of two numbers is equal to the square of their sum, and the numbers are within a certain range. The challenge is to find how many pairs of numbers satisfy the condition, such that both are within the range of $-10^{12}$ to $10^{12}$.
In mathematical terms, find the number of pairs of integers $(a, b)$ such that: $$a^3 + b^3 = (a + b)^2$$ Additionally, both $a$ and $b$ must be between $-10^{12}$ and $10^{12}$ (inclusive).
Express your answer as an integer in standard notation with no commas.

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