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A message signal with bandwidth $10 \; \mathrm{kHz}$ is Lower-Side Band SSB modulated with carrier frequency $f_{c 1}=10^{6} \mathrm{~Hz}$. The resulting signal is then passed through a Narrow-Band Frequency Modulator with carrier frequency $f_{c_{2}}=10^{9} \; \mathrm{Hz}$. The bandwidth of the output would be

1. $4 \times 10^{4} \mathrm{~Hz}$
2. $2 \times 10^{6} \mathrm{~Hz}$
3. $2 \times 10^{9} \mathrm{~Hz}$
4. $2 \times 10^{10} \mathrm{~Hz}$