There is one agent with utility function given by:
\begin{equation} U(c,l) = \frac{c^{1-\sigma}}{1-\sigma}-\frac{l^{1+\gamma}}{1+\gamma}\tag{1} \end{equation}
With budget constraint:
\begin{equation} c_{t} + k_{t+1} = (1+r_t)k_{t} + l_tw_t \tag{2} \end{equation}
The technology in the economy is a standard Cobb-Douglas with CRS:
\begin{equation} Y = K^{\alpha}L^{1-\alpha}\tag{3} \end{equation}
The FOC of the utility-maximization problem are:
\begin{align} c_{t+1} &= c_{t} [\beta (1+r_{t+1})] ^{\frac{1}{\sigma}}\tag{4}\\ % c_{t}^{\sigma} l_{t}^{\gamma} &= w\tag{5}\\ % \end{align}
While the FOCs of the profit-maximizing problem are:
$$r = \alpha K^{\alpha-1}L^{1-\alpha} \tag{6}$$
$$w = (1-\alpha) K^\alpha L^{-\alpha} \tag{7}$$
In the steady state, equation (4) implies the following
\begin{equation} r = \frac{1-\beta}{\beta}\tag{8} \end{equation}
Hence, to solve for the steady state, we need to solve for the following system of equation:
\begin{equation} C = rK + wL \tag{9} \end{equation}
\begin{equation} C^{\sigma} L^{\gamma} =w\tag{10} \end{equation}
$$r = \alpha K^{\alpha-1}L^{1-\alpha} \tag{11}$$
$$w = (1-\alpha) K^\alpha L^{-\alpha} \tag{12}$$
Where the unknowns are $C, K, L,w$ ($r$ is known).
But from here I can't solve for the exact values of these variables. What information I miss?