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The process for constructing a function that has a nested series of "for" loops with variable arguments would involve the following steps:

1. Define the function name and its input parameters, which should include the number of iterations for each for loop.

2. Create nested for loops inside the function. The number of for loops will be determined by the number of input parameters.

3. Inside the innermost for loop, include the desired code or operation that should be performed with the values of the variables from all the for loops.

4. Run the function with the desired values for the input parameters to test whether the code executes correctly.

5. Make any necessary adjustments or changes to the code until the desired outcome is achieved.

6. Use the function with various input values to further test its functionality and ensure that it works as intended.