Ridge vs. Lasso Regression

You are comparing two common regularization approaches for linear regression: Ridge ($L_2$) and Lasso ($L_1$). 1. Write down the optimization problem each method solves. What penalty term does each add to the ordinary least squares objective? 2. Ridge regression has a closed-form solution. Derive it. Why does Lasso not have one? 3. When would you prefer Ridge over Lasso, and vice versa? What does each method actually do to the coefficient vector $\hat{\beta}$?