Two Steps, and the Edge of Divergence

Gradient descent is run on the quadratic loss

$L(w) = 3w^2 - 12w + 7$

using the standard update

$w_{t+1} = w_t - \eta\, L'(w_t)$

starting from $w_0 = 5$ with a fixed learning rate $\eta = 0.25$.

Compute $w_2$, and determine the full range of $\eta > 0$ for which this iteration converges to the minimiser from any starting point.