Three Tiles, One Output

A one-pixel world. Every tile in this dataset is a single brightness value between $0$ and $1$, and the dataset contains exactly three kinds of tile:

A model is asked to produce a tile. The request never varies — it is always "a tile from this dataset" — so there is nothing to condition on: the model simply emits one number $\hat{y}$.

It is trained with squared error against a tile drawn at random from the dataset, so the quantity training actually drives down is the average squared error over the whole dataset:

$L(\hat{y}) = 0.25\,(\hat{y} - 0.0)^2 + 0.25\,(\hat{y} - 0.6)^2 + 0.50\,(\hat{y} - 1.0)^2$

Training runs long enough that $\hat{y}$ settles wherever $L$ is smallest.

What is $L$ at that point? Give the value to four decimal places.