\[ \LARGE \square^{\square^{\square^{\square}}} \]

You are given that the numbers \(1,2,3,4\) are to be filled in the square boxes as shown above (without repetition), forming an exponent towers? . Over all \(4!=24\) possible arrangements, let \(S\) be the minimal value that is achieved. How many arrangements of these numbers would produce the value of S?

**Details and Assumptions**

As an explicit example, a possible value of the resultant number is \(\large 2^{3^{1^4}} = 8 \).

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