\(n\text{-}\sc{PROVABILITY}_{\mathsf{T}}\ \) Given a formula \(\phi\) in the language of \(\mathsf{T}\) and a natural number \(n\), does there exist a valid \(\mathsf{T}\)-derivation \(\mathcal{D}\) of length \(\leq n\) such that \(\mathcal{D}\) is a valid proof of \(\phi\)?