Merging Intervals

You are given $N$ closed intervals $[l_i,r_i]$.

Choose two distinct intervals. Find the maximum possible length of their union.

The input is given from Standard Input in the following format:

$T$ $\mathrm{case}_1$ $\mathrm{case}_2$ $\vdots$ $\mathrm{case}_T$

Each case is given in the following format:

$N$ $l_1\ r_1$ $l_2\ r_2$ $\vdots$ $l_N\ r_N$

For each test case, print the maximum possible length of the union of two intervals on one line.

• $1 \leq T \leq 100$.
• $2 \leq N \leq 200\,000$.
• $0 \leq l_i \leq r_i \leq 10^9$ ($1 \leq i \leq N$).
• The sum of $N$ over all test cases does not exceed $200\,000$.