Regression through origin after reparametrisation

For a given data $(x_i, y_i)$, $i = 1, 2, \dots, n$, with $\sum_{i=1}^{n} x_i^2 > 0$, let $\hat{\beta}$ satisfy $\sum_{i=1}^{n} (y_i - \hat{\beta} x_i)^2 = \inf_{\beta \in \mathbb{R}} \sum_{i=1}^{n} (y_i - \beta x_i)^2$. Further, let $v_j = y_j - x_j$ and $u_j = 2x_j$, for $j = 1, 2, \dots, n$, and let $\hat{\gamma}$ satisfy $\sum_{i=1}^{n} (v_i - \hat{\gamma} u_i)^2 = \inf_{\gamma \in \mathbb{R}} \sum_{i=1}^{n} (v_i - \gamma u_i)^2$. If $\hat{\beta} = 10$, then the value of $\hat{\gamma}$ is