The maximum detection range of a radar system is modeled by $R_{max} = \\left( \\frac{P_t G^2 \\lambda^2 \\sigma}{(4\\pi)^3 P_{min}} \\right)^{1/4}$. If the transmitter power $P_t$ is doubled, the antenna gain $G$ is increased by $3$ dB (corresponding to a factor of 2), and the minimum detectable signal power $P_{min}$ is halved (with all other parameters constant), by what factor does the maximum range $R_{max}$ increase?