Buggy python submission (verdict: Time Limit Exceeded):
# coding: utf-8
import numpy as np


def solve(*args: str) -> str:
    n = int(args[0])
    LR = [tuple(map(int, a.split())) for a in args[1:]]
    L, R = zip(*LR)

    ret = 0

    p, q = np.argmax(L), np.argmin(R)
    lp, rq = L[p], R[q]
    if p == q:
        ret = max(0, 1+rq-lp) + max(0, 1+max(R)-min(L))
    else:
        AB = [[max(0, 1+r-lp), max(0, 1+rq-l)] for l, r in LR]
        AB.sort(key=lambda x: (x[0], -x[1]))
        A, B = zip(*AB)
        for i in range(1, n):
            ret = max(ret, min(A[i:]) + min(B[:i]))

    return str(ret)


if __name__ == "__main__":
    print(solve(*(open(0).read().splitlines())))


Fixed submission:
# coding: utf-8
import numpy as np


def solve(*args: str) -> str:
    n = int(args[0])
    LR = [tuple(map(int, a.split())) for a in args[1:]]
    L, R = zip(*LR)

    ret = 0

    p, q = np.argmax(L), np.argmin(R)
    lp, rq = L[p], R[q]

    ret = max(0, 1+rq-lp) + max(0, max(1+r-l for l, r in LR))

    AB = [[max(0, 1+r-lp), max(0, 1+rq-l)] for l, r in LR]
    AB.sort(key=lambda x: (x[0], -x[1]))
    A, B = map(list, zip(*AB))

    # for i in range(1, n):
    #     ret = max(ret, min(A[i:]) + min(B[:i]))
    b_min = 1+10**9
    for i in range(n-1):
        b_min = min(b_min, B[i])
        ret = max(ret, b_min + A[i+1])

    return str(ret)


if __name__ == "__main__":
    print(solve(*(open(0).read().splitlines())))

Buggy python submission (verdict: Runtime Error):
# coding: utf-8
from functools import lru_cache
import sys


def solve(*args: str) -> str:
    n = int(args[0])
    LR = []
    for a in args[1:]:
        LR.append(tuple(map(int, a.split())))

    dp = DP(LR)

    sys.setrecursionlimit(10**5)
    return str(dp.search_max(0, 0, 10**9, 0, 10**9))


class DP:
    def __init__(self, LR: list):
        self.LR = LR

    @lru_cache(maxsize=None)
    def search_max(self, index: int, l1: int, r1: int, l2: int, r2: int)\
            -> int:
        ret = 0

        if index < len(self.LR):
            l, r = self.LR[index]
            ret = max(
                self.search_max(index+1, max(l, l1), min(r, r1), l2, r2),
                self.search_max(index+1, l1, r1, max(l, l2), min(r, r2))
            )
        else:
            if l1 == 0 or l2 == 0:
                ret = 0
            else:
                ret = max(0, (r1-l1)) + max(0, (r2-l2) + 2)

        return ret


if __name__ == "__main__":
    n = input()
    print(solve(*([n]+[input() for _ in range(int(n))])))


Fixed submission:
# coding: utf-8
import numpy as np


def solve(*args: str) -> str:
    n = int(args[0])
    LR = [tuple(map(int, a.split())) for a in args[1:]]
    L, R = zip(*LR)

    ret = 0

    p, q = np.argmax(L), np.argmin(R)
    lp, rq = L[p], R[q]

    ret = max(0, 1+rq-lp) + max(0, max(1+r-l for l, r in LR))

    AB = [[max(0, 1+r-lp), max(0, 1+rq-l)] for l, r in LR]
    AB.sort(key=lambda x: (x[0], -x[1]))
    A, B = map(list, zip(*AB))

    # for i in range(1, n):
    #     ret = max(ret, min(A[i:]) + min(B[:i]))
    b_min = 1+10**9
    for i in range(n-1):
        b_min = min(b_min, B[i])
        ret = max(ret, b_min + A[i+1])

    return str(ret)


if __name__ == "__main__":
    print(solve(*(open(0).read().splitlines())))

Buggy python submission (verdict: Runtime Error):
# coding: utf-8
from functools import lru_cache


def solve(*args: str) -> str:
    n = int(args[0])
    LR = []
    for a in args[1:]:
        LR.append(tuple(map(int, a.split())))

    dp = DP(LR)

    return str(dp.search_max(0, 0, 10**9, 0, 10**9))


class DP:
    def __init__(self, LR: list):
        self.LR = LR

    @lru_cache(maxsize=None)
    def search_max(self, index: int, l1: int, r1: int, l2: int, r2: int)\
            -> int:
        ret = 0

        if index < len(self.LR):
            l, r = self.LR[index]
            ret = max(
                self.search_max(index+1, max(l, l1), min(r, r1), l2, r2),
                self.search_max(index+1, l1, r1, max(l, l2), min(r, r2))
            )
        else:
            if l1 == 0 or l2 == 0:
                ret = 0
            else:
                ret = max(0, (r1-l1)) + max(0, (r2-l2) + 2)
        # print((l1, r1, l2, r2))
        return ret


if __name__ == "__main__":
    n = input()
    print(solve(*([n]+[input() for _ in range(int(n))])))


Fixed submission:
# coding: utf-8
import numpy as np


def solve(*args: str) -> str:
    n = int(args[0])
    LR = [tuple(map(int, a.split())) for a in args[1:]]
    L, R = zip(*LR)

    ret = 0

    p, q = np.argmax(L), np.argmin(R)
    lp, rq = L[p], R[q]

    ret = max(0, 1+rq-lp) + max(0, max(1+r-l for l, r in LR))

    AB = [[max(0, 1+r-lp), max(0, 1+rq-l)] for l, r in LR]
    AB.sort(key=lambda x: (x[0], -x[1]))
    A, B = map(list, zip(*AB))

    # for i in range(1, n):
    #     ret = max(ret, min(A[i:]) + min(B[:i]))
    b_min = 1+10**9
    for i in range(n-1):
        b_min = min(b_min, B[i])
        ret = max(ret, b_min + A[i+1])

    return str(ret)


if __name__ == "__main__":
    print(solve(*(open(0).read().splitlines())))