A student investigates a high-pressure air-treatment reactor that oxidizes \(\text{NO(g)}\) according to the following proposed mechanism. The first elementary step is a rapid pre-equilibrium.

\[ 2\,\text{NO(g)} \rightleftharpoons \text{N}_2\text{O}_2\text{(g)} \]

The second elementary step is slow.

\[ \text{N}_2\text{O}_2\text{(g)}+\text{O}_2\text{(g)}\rightarrow 2\,\text{NO}_2\text{(g)} \]

At \(298\ \text{K}\), the rapid step has \(K_c=0.20\ \text{M}^{-1}\), and the slow step has \(k_2=1.0\times10^3\ \text{M}^{-1}\text{s}^{-1}\). Immediately after the pre-equilibrium is established, \([\text{NO}]=0.050\ \text{M}\) and \([\text{O}_2]=0.100\ \text{M}\). What initial rate of formation of \(\text{NO}_2\) is predicted by the mechanism?