Matrices And Determinants

Let \(P\) be a square matrix such that \(P^{2}=I-P\). For \(\alpha, \beta, \gamma, \delta \in \mathbb{N}\), if \(P^{\alpha}+P^{\beta}=\gamma I-29 P\) and \(P^{\alpha}-P^{\beta}=\delta I-13 P\), then \(\alpha+\beta+\gamma-\delta\) is equal to :