An angle modulated signal is given by

${}_{EM}^{}(t)=4cos(2{10}^{6}t+2sin(2{10}^{3}t)+4sin(4{10}^{3}t)).$

$(1)$ What is the bandwidth of the angle modulated signal? $($Carson$\text{'}$s rule$)$

$(2)$ If the angle modulated signal is an FM signal with a frequency deviation constant of $4{10}^{3}ra\frac{d}{s\frac{ec}{\text{v}\text{o}\text{l}\text{t}}},$ determine the message signal $m(t).$

$(3)$ What is the value of the frequency deviation ratio $($i$.$e$.$ modulation index$)?$