Score : $300$ points

Problem Statement

Snuke has two number sequences $a$ and $b$, each of length $N$. The $i$-th number of $a$ is $a_i$, and that of $b$ is $b_i$.

Using these sequences, he has decided to make a new sequence $c$ of length $N$. For each $n$ such that $1 \leq n \leq N$, $c_n$, the $n$-th number of $c$, is the maximum value of $a_i b_j$ for a pair $(i,j)$ such that $1 \leq i \leq j \leq n$. Formally, we have $c_n = \max_{1 \leq i \leq j \leq n} a_{i}b_{j}$.

Find $c_1, c_2, \ldots, c_{N}$.

Constraints

• All values in input are integers.
• $1 \leq N \leq 2 \times 10^{5}$
• $1 \leq a_i, b_i \leq 10^9$

Input

Input is given from Standard Input in the following format:

$N$

$a_{1}$ $a_{2}$ $\cdots$ $a_{N}$

$b_{1}$ $b_{2}$ $\cdots$ $b_{N}$

Print

Print $N$ lines. The $n$-th line from the top should contain $c_{n}$.

3
3 2 20
1 4 1

3
12
20

We have:

• $c_{1} = \max(a_{1}b_{1}) = 3$;
• $c_{2} = \max(a_{1}b_{1}, a_{1}b_{2},a_{2}b_{2}) = 12$;
• $c_{3} = \max(a_{1}b_{1}, a_{1}b_{2}, a_{1}b_{3}, a_{2}b_{2},a_{2}b_{3},a_{3}b_{3}) = 20$.

20
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974365976 488724815 821047998 371238977 256373343 218153590 546189624 322430037 131351929 768434809 253508808 935670831 251537597 834352123 337485668 272645651 61421502 439773410 621070911 578006919

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