Let $$

be a boolean formula in variables $$

and $$

$,$ of any length, using any of the logical operators $,,,,$

$.$ For example, $$

could be $(())$

$.$ Prove that $$

is logically equivalent to a formula of the form

$$

where $$

is one of $,,$

$,$ and $$

and $$

are chosen from $\{,,,\}$

$.$

Note: you do not need to find an "elegant" proof of this.