Problem:
Score : 300 points

Problem Statement

The season for Snuke Festival has come again this year. First of all, Ringo will perform a ritual to summon Snuke. For the ritual, he needs an altar, which consists of three parts, one in each of the three categories: upper, middle and lower.

He has N parts for each of the three categories. The size of the i-th upper part is A_i, the size of the i-th middle part is B_i, and the size of the i-th lower part is C_i.

To build an altar, the size of the middle part must be strictly greater than that of the upper part, and the size of the lower part must be strictly greater than that of the middle part. On the other hand, any three parts that satisfy these conditions can be combined to form an altar.

How many different altars can Ringo build? Here, two altars are considered different when at least one of the three parts used is different.

Constraints

1 \leq N \leq 10^5

1 \leq A_i \leq 10^9(1\leq i\leq N)

1 \leq B_i \leq 10^9(1\leq i\leq N)

1 \leq C_i \leq 10^9(1\leq i\leq N)

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 ... A_N
B_1 ... B_N
C_1 ... C_N

Output

Print the number of different altars that Ringo can build.

Sample Input 1

2
1 5
2 4
3 6

Sample Output 1

3

The following three altars can be built:

Upper: 1-st part, Middle: 1-st part, Lower: 1-st part

Upper: 1-st part, Middle: 1-st part, Lower: 2-nd part

Upper: 1-st part, Middle: 2-nd part, Lower: 2-nd part

Sample Input 2

3
1 1 1
2 2 2
3 3 3

Sample Output 2

27

Sample Input 3

6
3 14 159 2 6 53
58 9 79 323 84 6
2643 383 2 79 50 288

Sample Output 3

87

Solution in LuaJIT (2.0.4) (judge verdict: Wrong Answer)

function calculate_pattern(parts, i)
    local under_pointer = 0
    for p = 1, #parts[i] do
        while under_pointer + 1 <= #parts[i+1] and parts[i+1][under_pointer+1].size > parts[i][p].size do
            under_pointer = under_pointer + 1
        end
        if under_pointer > 0 then
            parts[i][p].pattern = pattern_sum(parts, i+1, under_pointer)
        else
            parts[i][p].pattern = 0
        end
    end
end

pattern_sum = nil
do
    local memo = {}
    for i = 1, 3 do
        table.insert(memo, {})
    end
    function pattern_sum(parts, i, p_max)
        if memo[i][p_max] then
            return memo[i][p_max]
        end
        
        local sum = 0
        for p = 1, p_max do
            sum = sum + parts[i][p].pattern
        end
        memo[i][p_max] = sum
        return sum
    end
end

function main()
    local inputs = {}
    for i = 1, 4 do
        table.insert(inputs, io.read())
    end
    
    local n = tonumber(inputs[1])
    local parts = {}
    for i = 1, 3 do
        table.insert(parts, {})
        for size in inputs[1+i]:gmatch("%d+") do
            table.insert(parts[i], {size = tonumber(size), pattern = 1})
        end
    end
    
    for i = 1, #parts do
        table.sort(parts[i], function(a, b) return b.size < a.size end)
    end
    
    for i = #parts - 1, 1, -1 do
        calculate_pattern(parts, i)
    end
    
    print(pattern_sum(parts, 1, #parts[1]))
end

main()