Square-Cube Digit Usage The point of this challenge is to find the smallest positive integer that uses up at least all of the digits supplied in the input after it is squared and cubed.

So, when provided an input such as 0123456789 (i.e. a number that finds the result):

69² = 4761
69³ = 328509


It means that 69 is the result of such an input. (Fun fact: 69 is the smallest number that uses up all of the decimal digits of 0 to 9 after squaring and cubing.)

Specification

The input doesn't have to be unique. For example, the input can be 1466 and here's the result:

4² = 16
4³ = 64


That means we can't just fulfill this input by just doing a number that only uses 1 digit of 6, it has to have 2 digits of 6 in the output.

Test cases

Here's an exhaustive list of all numbers from 1 to 10000.

1333 (or 3133/3313/3331) -&gt; 111


Input specification


You may take input as a list of digits.
Since the test cases (somehow) have a bug in it, here's an extra rule: the 0's in the input will be ignored.
05AB1E, 14 12 13 bytes
-2 bytes thanks to @KevinCruijssen
+1 byte thanks to @Grimmy
∞.Δ23SmJœIÅ?Z

Try it online!

Explanation
∞.Δ                  - First number that... 
   23Sm              - Power of 2 and 3 [n^2, n^3] 
       J             - Concatenated
        œ            - Permutations of this number
         IÅ?Z        - any of these start with the number 

By the way it's quite slow...