def r_modulo_closed_form(n, m):
    # From e132.py
    assert n > 0 and m > 0
    return ((pow(10, n, 9 * m) - 1) // 9) % m


def s(n, mod):
    a = n % 9
    b = n // 9
    r = (a + 1) * pow(10, b, mod) - 1
    return r % mod


def s_big(n, mod):
    # s = (n % 9 + 1) * 10^(n/9) - 1
    # S = sum(k=0, n/9) (1 * 10^k + 2 * 10^k + ... + 9 * 10^k) - n
    #   = 45 * sum(k=0, n/9) 10^k - n
    #   = 45 * r(n/9) - n
    # Plus some extra math to make n % 9 == 8 so that the above works.

    r = -1
    while n % 9 != 8:
        n += 1
        r -= s(n, mod)

    r += (r_modulo_closed_form(n // 9 + 1, mod) * 45) % mod
    r -= n
    return r % mod


def s_big_naiv(n, mod):
    r = 0
    for i in range(1, n + 1):
        r += s(i, mod)
        if mod is not None:
            r %= mod
    return r


def euler_684():
    assert s_big_naiv(20, 11) == s_big(20, 11)
    r = 0
    mod = 1_000_000_007
    a, b = 0, 1
    for _ in range(2, 91):
        a, b = b, a + b
        r += s_big(b, mod)
    return r % mod


if __name__ == "__main__":
    solution = euler_684()
    print("e684.py: " + str(solution))
    assert(solution == 922058210)