Substitute each capital letter in bold by a different base ten digit from 0 to 9, such that (TAU)^.BETA when rounded off to the nearest integer is equal to PI, and the absolute difference of (TAU)^.BETA and PI is the minimum. Each of A, P and T is nonzero.

Notes:

(i) PI does not represent the quantity π and, .BETA denotes a decimal fraction.

(ii) Relevant alphametic rules are applicable for this problem.