PE 0012: Highly Divisible Triangular Number

The sequence of triangle numbers is generated by adding the natural numbers. So the $7$^th triangle number would be $1+2+3+4+5+6+7=28$. The first ten terms would be:

$$1,3,6,10,15,21,28,36,45,55,\dots$$

Let us list the factors of the first seven triangle numbers:

\begin{align} \mathbf{1} &\colon 1\ \mathbf{3} &\colon 1,3\ \mathbf{6} &\colon 1,2,3,6\ \mathbf{10} &\colon 1,2,5,10\ \mathbf{15} &\colon 1,3,5,15\ \mathbf{21} &\colon 1,3,7,21\ \mathbf{28} &\colon 1,2,4,7,14,28 \end

We can see that $28$ is the first triangle number to have over five divisors.

What is the value of the first triangle number to have over five hundred divisors?

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