Assume that we have a \(\mathrm{CO}_{2}\) laser that has a bandwidth of \(\Delta v=\) \(1 \mathrm{GHz}\) at \(=10.6 \mu \mathrm{m}\). For simplicity, the spectral profile is assumed to be rectangular. The length of the cavity is equal to \(1 \mathrm{~m}\). Calculate the number of longitudinal modes that can oscillate in this cavity. At what distance must the cavity mirror be placed so that at least one of the modes falls in the amplification band?