USACO 2024 February Contest, Platinum Problem 1. Lazy Cow - Problem

Description

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

The first line contains $D$. The $i$th of the next $D$ lines contains two space-separated integers $m_i$ and $b_i$.

Output

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

Constraints

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.

Sample 1 Input

Sample 1 Output

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.

For each $i$, it can be shown that Bessie cannot satisfy the first $i$ demands using less energy.

Sample 2 Input

2
100 5
100 1000000000000

Sample 2 Output

5
627323485

Sample 3 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 3 Output

10440: USACO 2024 February Contest, Platinum Problem 1. Lazy Cow

Description

Input

Output

Constraints

Sample 1 Input

Sample 1 Output

Sample 2 Input

Sample 2 Output

Sample 3 Input

Sample 3 Output

HINT

Source/Category

10440: USACO 2024 February Contest, Platinum Problem 1. Lazy Cow

Description

Input

Output

Constraints

Sample 1 Input Copy

Sample 1 Output Copy

Sample 2 Input Copy

Sample 2 Output Copy

Sample 3 Input Copy

Sample 3 Output Copy

HINT

Source/Category

Sample 1 Input

Sample 1 Output

Sample 2 Input

Sample 2 Output

Sample 3 Input

Sample 3 Output