Problem Statement

Mountaineers Mr. Takahashi and Mr. Aoki recently trekked across a certain famous mountain range. The mountain range consists of $N$ mountains, extending from west to east in a straight line as Mt. $1$, Mt. $2$, ..., Mt. $N$. Mr. Takahashi traversed the range from the west and Mr. Aoki from the east.

The height of Mt. $i$ is $h_i$, but they have forgotten the value of each $h_i$. Instead, for each $i$ ($1 ≤ i ≤ N$), they recorded the maximum height of the mountains climbed up to the time they reached the peak of Mt. $i$ (including Mt. $i$). Mr. Takahashi's record is $T_i$ and Mr. Aoki's record is $A_i$.

We know that the height of each mountain $h_i$ is a positive integer. Compute the number of the possible sequences of the mountains' heights, modulo $10^9 + 7$.

Note that the records may be incorrect and thus there may be no possible sequence of the mountains' heights. In such a case, output $0$.

Constraints

• $1 ≤ N ≤ 10^5$
• $1 ≤ T_i ≤ 10^9$
• $1 ≤ A_i ≤ 10^9$
• $T_i ≤ T_{i+1}$ ($1 ≤ i ≤ N - 1$)
• $A_i ≥ A_{i+1}$ ($1 ≤ i ≤ N - 1$)

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Input

The input is given from Standard Input in the following format:

$N$ $T_1$ $T_2$ $...$ $T_N$ $A_1$ $A_2$ $...$ $A_N$

Output

Print the number of possible sequences of the mountains' heights, modulo $10^9 + 7$.

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Sample Input 1

5
1 3 3 3 3
3 3 2 2 2

Sample Output 1

4

The possible sequences of the mountains' heights are:

• $1, 3, 2, 2, 2$
• $1, 3, 2, 1, 2$
• $1, 3, 1, 2, 2$
• $1, 3, 1, 1, 2$

for a total of four sequences.

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Sample Input 2

5
1 1 1 2 2
3 2 1 1 1

Sample Output 2

0

The records are contradictory, since Mr. Takahashi recorded $2$ as the highest peak after climbing all the mountains but Mr. Aoki recorded $3$.

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Sample Input 3

10
1 3776 3776 8848 8848 8848 8848 8848 8848 8848
8848 8848 8848 8848 8848 8848 8848 8848 3776 5

Sample Output 3

884111967

Don't forget to compute the number modulo $10^9 + 7$.

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Sample Input 4

1
17
17

Sample Output 4

1

Some mountain ranges consist of only one mountain.