There is more to confession than reciting the contrition
The Act of Contrition is not even a part of confession. It is frequently a part of penance following absolution.
The confession is like this. First, the faithful examines his conscience with himself and attempts to identify where he has yet unconfessed sin. Familiar lists of sins, such as the Ten Commandments, or the so-called cardinal sins, help in the analysis but either method of self-examination, or no particular method at all, can be used. Next, he tells his sins to the priest formally (the formal setting guarantees that priest is bound by the absolute privacy of the confessional). If the faithful has not been able to determine during the examination of conscience that he has sinned, but suspects that he might, he discusses that with the priest. Then the priest prayerfully decides to forgive or retain the sin. If the sins are forgiven, a prayer is said to that effect and the sins are then absolved. Then a penance is given, usually a series of prayers or a charitable act, and the confession conccludes at this point