Problem Statement

Each individual of a certain organism is assigned, at the moment of its birth, an integer value among $1, \dots, N$ as its individual value. Also, exactly $1$ day after its birth, it gives birth to $0$ or more individuals, and at the same time it disappears.

The individual value of a newborn individual and the number of individuals that $1$ individual gives birth to are determined probabilistically as follows, independently of other individuals:

• the probability that the individual value of a newborn individual is $i\ (=1,\dots,N)$ is $p_i$,
• the probability that the number of individuals born from $1$ individual with individual value $i$ is $k\ (=0,1,\dots)$ is $(1-q_i) q_i^k$.

Now, $A_1,\dots,A_N$ individuals with individual values $1,\dots,N$, respectively, have just been born. No other individuals exist. Find the probability $P$ that the organism is extinct after $10^{10^{100}}$ days, that is, that the number of individuals is $0$.

Since $P$ may be very small, output the value of its natural logarithm $\log P$. Under the constraints of this problem, it can be shown that $P > 0$.

Input

The input is given in the following format:

$N$
$p_1 \quad p_2 \quad \cdots \quad p_N$
$q_1 \quad q_2 \quad \cdots \quad q_N$
$A_1 \quad A_2 \quad \cdots \quad A_N$

• $N,\ A_1,\dots,A_N$ are integers
• $p_1,\dots,p_N,\ q_1,\dots,q_N$ are decimals with $6$ digits after the decimal point (padded with $0$ so as to have $6$ digits)
• $1 \leq N \leq 10^5$
• $0 < p_i \leq 1 \quad (1 \leq i \leq N)$
• $\sum_{i=1}^N p_i = 1$
• $\color{red}{0.6 \leq q_i \leq 0.9} \quad (1 \leq i \leq N)$
• $0 \leq A_i \leq 10^4 \quad (1 \leq i \leq N)$

Note: The somewhat unnatural-looking constraint on $q_i$ is there so that solutions using the double type can be accepted without much concern about numerical errors.

Output

For the probability $P$ that the organism is extinct after $10^{10^{100}}$ days, output the value of the natural logarithm $\log P$. Output a newline at the end.

Your answer is considered correct if its absolute or relative error from the expected answer is at most $10^{-5}$.

Sample

1
1.000000
0.600000
1

-0.4054651081

2
0.900000 0.100000
0.900000 0.900000
0 0

0

6
0.332343 0.068862 0.076451 0.120693 0.263170 0.138481
0.839426 0.728493 0.810372 0.839476 0.827081 0.890719
7481 5029 9181 6723 8291 9836

-75700.00551