What is the form of a verb after "achieve" [closed]
Achieve takes a noun phrase as a direct object. If a verbal form is used it should take a gerund as a direct object, not an infinitive.
I have achieved playing piano well.
There is no simple rule for telling which verbs take which verbal forms as an object; some take gerunds, some take infinitives, and some can take either (OWL). The trend is that stative verbs (like "achieve") are more likely to take a gerund, and action verbs are more likely to use an infinitive, but there are many exceptions (TFD).