## USACO 2024 February Contest, Platinum

## Problem 1. Lazy Cow

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Bessie is hard at work preparing test cases for the USA Cowmputing Olympiad February contest. Each minute, she can choose to not prepare any tests, expending no energy; or expend $3^{a-1}$ energy preparing $a$ test cases, for some positive integer $a$.

Farmer John has $D$ ($1\le D\le 2\cdot 10^5$) demands. For the $i$th demand, he tells Bessie that within the first $m_i$ minutes, she needs to have prepared at least $b_i$ test cases in total ($1\le m_i\le 10^6, 1 \leq b_i \leq 10^{12}$).

Let $e_i$ be the smallest amount of energy Bessie needs to spend to satisfy the first $i$ demands. Print $e_1,\dots,e_D$ modulo $10^9+7$.

#### INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $D$. The $i$th of the next $D$ lines contains two space-separated integers $m_i$ and $b_i$.#### OUTPUT FORMAT (print output to the terminal / stdout):

Output $D$ lines, the $i$th containing $e_i \text{ mod } 10^9+7$.#### SAMPLE INPUT:

4 5 11 6 10 10 15 10 30

#### SAMPLE OUTPUT:

21 21 25 90

For the first test case,

- $i=1$: If Bessie creates $[2, 3, 2, 2, 2]$ test cases on the first $5$ days, respectively, she would have expended $3^1 + 3^2 + 3^1 + 3^1 + 3^1 = 21$ units of energy and created $11$ test cases by the end of day $5$.
- $i=2$: Bessie can follow the above strategy to ensure $11$ test cases are created by the end of day $5$, and this will automatically satisfy the second demand.
- $i=3$: If Bessie creates $[2, 3, 2, 2, 2, 0, 1, 1, 1, 1]$ test cases on the first $10$ days, respectively, she would have expended $25$ units of energy and satisfied all demands. It can be shown that she cannot expend less energy.
- $i=4$: If Bessie creates 3 test cases on each of the first $10$ days she would have expended $3^{2}\cdot 10 = 90$ units of energy and satisfied all demands.

#### SAMPLE INPUT:

2 100 5 100 1000000000000

#### SAMPLE OUTPUT:

5 627323485

#### SAMPLE INPUT:

20 303590 482848034083 180190 112716918480 312298 258438719980 671877 605558355401 662137 440411075067 257593 261569032231 766172 268433874550 8114 905639446594 209577 11155741818 227183 874665904430 896141 55422874585 728247 456681845046 193800 632739601224 443005 623200306681 330325 955479269245 377303 177279745225 880246 22559233849 58084 155169139314 813702 758370488574 929760 785245728062

#### SAMPLE OUTPUT:

108753959 108753959 108753959 148189797 148189797 148189797 148189797 32884410 32884410 32884410 32884410 32884410 32884410 32884410 3883759 3883759 3883759 3883759 3883759 3883759

#### SCORING:

- Inputs 4-5: $D\le 100$ and $m_i \le 100$ for all $i$
- Inputs 6-8: $D\le 3000$
- Inputs 9-20: No additional constraints.

Problem credits: Brandon Wang and Claire Zhang