Idioms "as a means to " vs "as a means for"

Hi

As per my knowledge , both the idioms are correct and both somehow mean the same thing ie a method for achieving something

Examples
As a means to
1 - cities are stressing the arts as a means to greater economic development
2 - They sought to increase the size of the available labor force as a means to keep men’s wages low.

As a means for
1 - Parents should participate in the program as a means of decreasing the cost for their children’s college education.

2 -None of choices indicates that the ban is ill chosen as a means of reaching the town council’s goals

My Question - How is their usage different ? Is “as means of” always followed by a ingVerb( present participle) .

Notice that the gerund follows the ‘of’ form.