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E - Souvenir

Score : $500$ points

Problem Statement

There are $N$ cities. There are also one-way direct flights that connect different cities.
The availability of direct flights is represented by $N$ strings $S_1,S_2,\ldots,S_N$ of length $N$ each. If the $j$-th character of $S_i$ is Y, there is a direct flight from city $i$ to city $j$; if it is N, there is not.
Each city sells a souvenir; city $i$ sells a souvenir of value $A_i$.

Consider the following problem:

Takahashi is currently at city $S$ and wants to get to city $T$ (that is different from city $S$) using some direct flights.
Every time he visits a city (including $S$ and $T$), he buys a souvenir there.
If there are multiple routes from city $S$ to city $T$, Takahashi decides the route as follows:

• He tries to minimize the number of direct flights in the route from city $S$ to city $T$.
• Then he tries to maximize the total value of the souvenirs he buys.

Determine if he can travel from city $S$ to city $T$ using the direct flights. If he can, find the "number of direct flights" and "total value of souvenirs" in the route that satisfies the conditions above.

You are given $Q$ pairs $(U_i,V_i)$ of distinct cities.
For each $1\leq i\leq Q$, print the answer to the problem above when $S=U_i$ and $T=V_i$.

Constraints

• $2 \leq N \leq 300$
• $1\leq A_i\leq 10^9$
• $S_i$ is a string of length $N$ consisting of Y and N.
• The $i$-th character of $S_i$ is N.
• $1\leq Q\leq N(N-1)$
• $1\leq U_i,V_i\leq N$
• $U_i\neq V_i$
• If $i \neq j$, then $(U_i,V_i)\neq (U_j,V_J)$.
• $N,A_i,Q,U_i$, and $V_i$ are all integers.

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Input

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

$N$

$A_1$ $A_2$ $\ldots$ $A_N$

$S_1$

$S_2$

$\vdots$

$S_N$

$Q$

$U_1$ $V_1$

$U_2$ $V_2$

$\vdots$

$U_Q$ $V_Q$

Output

Print $Q$ lines.
The $i$-th $(1\leq i\leq Q)$ line should contain Impossible if he cannot travel from city $U_i$ to city $V_i$; if he can, the line should contain "the number of direct flights" and "the total value of souvenirs" in the route chosen as described above, in this order, separated by a space.

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5
30 50 70 20 60
NYYNN
NNYNN
NNNYY
YNNNN
YNNNN
3
1 3
3 1
4 5

1 100
2 160
3 180

For $(S,T)=(U_1,V_1)=(1,3)$, there is a direct flight from city $1$ to city $3$, so the minimum possible number of direct flights is $1$, which is achieved when he uses that direct flight. In this case, the total value of the souvenirs is $A_1+A_3=30+70=100$.

For $(S,T)=(U_2,V_2)=(3,1)$, the minimum possible number of direct flights is $2$. The following two routes achieve the minimum: cities $3\to 4\to 1$, and cities $3\to 5\to 1$. Since the total value of souvenirs in the two routes are $70+20+30=120$ and $70+60+30=160$, respectively, so he chooses the latter route, and the total value of the souvenirs is $160$.

For $(S,T)=(U_3,V_3)=(4,5)$, the number of direct flights is minimum when he travels cities $4\to 1\to 3\to 5$, where the total value of the souvenirs is $20+30+70+60=180$.

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2
100 100
NN
NN
1
1 2

Impossible

There may be no direct flight at all.