The step $\boldsymbol{\beta}^{t+1}$ in solving an optimization problem is given below.
$$\boldsymbol{\beta}^{t+1}=\mathrm{argmin}_{\boldsymbol{\beta}}\frac{1}{2}\left\Vert \mathbf{u}^{t}-\boldsymbol{\alpha}^{t+1}+\mathbf{r}-\mathbf{A}\boldsymbol{\beta}\right\Vert _{2}^{2}+\frac{1}{\rho}\left\Vert \boldsymbol{\beta}\right\Vert _{1}$$
The source stated that with a linearization in $\boldsymbol{\beta}=\boldsymbol{\beta}^{t+1}$, the expression can be written as $$\boldsymbol{\beta}^{t+1}=\mathrm{argmin}_{\boldsymbol{\beta}}\frac{1}{2}\left\Vert \boldsymbol{\beta}-\boldsymbol{\beta}^{t}+\mathbf{A}^{T}\left(\mathbf{A}\boldsymbol{\beta}^{t}-\mathbf{u}^{t}+\boldsymbol{\alpha}^{t+1}-\mathbf{r}\right)/\gamma\right\Vert _{2}^{2}+\frac{1}{\rho\gamma}\left\Vert \boldsymbol{\beta}\right\Vert _{1}$$ where $\gamma=\left\Vert \mathbf{A}\right\Vert _{2}^{2}$. Here, $\left\Vert \mathbf{A}\right\Vert _{2}$ to denote the largest singular value of $\mathbf{A}$. Can you help me with how this statement was obtained?
(source, Equations (3) and (8).)