Well-foundedness proof for ${\mathrm{\Pi }}_1^1$-reflection

In the lecture notes it is shown that an ordinal ${\psi }_{\mathrm{\Omega }}({\epsilon }_{{\mathbb{S}}^{+}+1})$ is an upper bound for the proof-theoretic ordinal of a set theory $\mathsf{K}\mathsf{P}\omega +(M{\prec }_{{\mathrm{\Sigma }}_1}V)$. In this note we show that $\mathsf{K}\mathsf{P}\omega +(M{\prec }_{{\mathrm{\Sigma }}_1}V)$ proves the well-foundedness up to ${\psi }_{\mathrm{\Omega }}({\omega }_n({\mathbb{S}}^{+}+1))$ for each $n$.