Maximum Detour

The geometric graph $G = (V, E)$ in this problem is a path of $N$ vertices $v_1, v_2, \ldots, v_N$, with one edge joining $v_i$ and $v_{i+1}$ for every $i$ with $1 \le i \le N-1$. Each vertex is a point on the plane, and each edge is the straight segment between the two points it joins.

Read $G$ as a road network you may travel only along its edges. Getting from one vertex to another is then usually longer than the straight line between them, and the detour is the ratio of the two. For two vertices $v_i$ and $v_j$ with $i < j$,

$D(G, v_i, v_j) = d_G(v_i, v_j) / d(v_i, v_j)$

where $d(p, q)$ is the Euclidean distance between two points $p$ and $q$, and $d_G(v_i, v_j) = d(v_i, v_{i+1}) + d(v_{i+1}, v_{i+2}) + \ldots + d(v_{j-1}, v_j)$ is the length of the walk along the path. The maximum detour $D(G)$ is the largest detour over all pairs of distinct vertices of $G$.

The figure shows a geometric graph $G$ that is a path of 7 vertices $v_1, \ldots, v_7$. The numbers are Euclidean distances between pairs of vertices, so $d(v_1, v_2) = 10$ and $d(v_1, v_4) = 4$. Here $D(G, v_1, v_4) = d_G(v_1, v_4) / d(v_1, v_4) = (10 + 10 + 9) / 4 = 29 / 4$.

You are given the $N$ points of one such path in order. Compute the maximum detour $D(G)$.

The first line has the number of test cases $T$ ($1 \le T \le 20$). Each test case begins with a line holding the number of vertices $N$ ($2 \le N \le 10000$), followed by $N$ lines holding the coordinates of $v_1, v_2, \ldots, v_N$ in that order, one vertex per line. The two integers on a line are separated by one space. Every coordinate is an integer between $-10000$ and $10000$. No two consecutive vertices are the same point, so every edge has positive length.

Print one line for each test case.

If $D(G)$ is at least 1000, print TOO LARGE. Otherwise print $D(G)$ rounded to two digits after the decimal point, always writing both digits: print 1.00, not 1.

The path may cross itself. Two vertices that are not neighbours may even lie on the same point, and then the distance between them is 0 and $D(G)$ grows without bound, so the answer is TOO LARGE.

In every test case the exact value of $D(G)$ is farther than $10^{-6}$ from 1000, and farther than $10^{-6}$ from any value halfway between two multiples of 0.01 such as 0.005, 0.015 or 0.025. The rounding direction is never ambiguous.