2026 AIME I Problem 6

The product of all positive real numbers x satisfying the equation \sqrt[20]{x^{\log_{2026}x}}=26x is an integer P. Find the number of positive integer divisors of P.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.