Catching Shade in Flatland

It is always sunny in Flatland, which gets annoying, so Alex heads to a park to catch some shade.

The park is a square of side $400$ centred at the origin of the Cartesian plane, with vertices at $(\pm 200, \pm 200)$. A point-sized bench sits at the centre of the park (the origin), where Alex sits. The Sun is a point that travels clockwise along the circle centred at the origin with radius $500$. At midnight the Sun is at $(0, 500)$, and it takes $24$ hours to complete one full revolution.

Several trees in the park provide shade. Each tree is a solid circle with integer centre coordinates and integer radius. Consider the ray of sunlight that travels from the Sun to the bench. Its shade value at a given moment is the sum, over all trees, of the length of the chord cut from each tree by that ray — that is, the total length of tree the sunlight must pass through on its way from the Sun to the bench. (A tree lying beyond the bench, on the far side from the Sun, does not block this sunlight and contributes nothing.)

We measure the shade value once every minute, on the minute, over a $24$-hour period starting at midnight: at 00:00, 00:01, ..., 23:59 ($1440$ measurements). Determine the maximum of these shade values.

The input contains several test cases (at most $100$). Each test case begins with a line containing an integer $N$ ($1 \le N \le 200$), the number of trees. Each of the next $N$ lines contains three integers $x_i$, $y_i$, $r_i$ — the centre coordinates and radius of the $i$-th tree.

The last test case is followed by a line containing a single $0$.

No tree contains or intersects another (two trees may touch), and no tree extends outside the park. No tree contains or touches the bench.

For each test case, print on its own line the maximum shade value, rounded to exactly $3$ decimal places.