Problem 732

Standing on the shoulders of trolls

$N$ trolls are in a hole that is $D_N$ cm deep. The $n$-th troll is characterized by:

the distance from his feet to his shoulders in cm, $h_n$
the length of his arms in cm, $l_n$
his IQ (Irascibility Quotient), $q_n$.

Trolls can pile up on top of each other, with each troll standing on the shoulders of the one below him. A troll can climb out of the hole and escape if his hands can reach to the surface. Once a troll escapes he cannot participate any further in the escaping effort.

The trolls execute an optimal strategy for maximizing the total IQ of the escaping trolls, defined as $Q(N)$.

Let
$r_n = \left[ \left( 5^n \bmod (10^9 + 7) \right) \bmod 101 \right] + 50$
$h_n = r_{3n}$
$l_n = r_{3n+1}$
$q_n = r_{3n+2}$
$D_N = \frac{1}{\sqrt{2}} \sum_{n=0}^{N-1} h_n$.

For example, the first troll ($n=0$) is $51$cm tall to his shoulders, has $55$cm long arms, and has an IQ of $75$.

You are given that $Q(5) = 401$ and $Q(15)=941$.

Find $Q(1000)$.

站在巨魔的肩膀上

$N$只巨魔（编号为$0$至$N-1$）掉进了深度为$D_N$厘米的洞中，其中编号为$n$的巨魔有三项特征：

他的肩高为$h_n$厘米；
他的臂长为$l_n$厘米；
他的怒商（怒气商数，简称IQ）为$q_n$。

巨魔们可以通过站在别的巨魔的肩膀上来搭起人梯，只要人梯最上方的巨魔的手臂能够够到地表，这只巨魔就能逃生。已经逃生的巨魔不能回洞中继续帮助其他巨魔逃生。

假设巨魔采用了能够使得所有逃出巨魔的IQ之和最高的策略，并记该策略下IQ之和为$Q(N)$。

记
$r_n = \left[ \left( 5^n \bmod (10^9 + 7) \right) \bmod 101 \right] + 50$
$h_n = r_{3n}$
$l_n = r_{3n+1}$
$q_n = r_{3n+2}$
$D_N = \frac{1}{\sqrt{2}} \sum_{n=0}^{N-1} h_n$.

例如，第一只（编号为$n=0$）巨魔的肩高为$51$厘米，臂长为$55$厘米，IQ为$75$。

已知$Q(5) = 401$，$Q(15)=941$。

求$Q(1000)$。