Score : $500$ points

Problem Statement

For a sequence $P = (P_1, \ldots, P_N)$, let $\mathrm{LIS}(P)$ denote the length of a longest increasing subsequence.

You are given permutations $A = (A_1, \ldots, A_N)$ and $B = (B_1, \ldots, B_N)$ of integers from $1$ through $N$. You may perform the following operation on these sequences any number of times (possibly zero).

• Choose an integer $i$ such that $1\leq i\leq N-1$. Swap $A_i$ and $A_{i+1}$, and swap $B_i$ and $B_{i+1}$.

Find the maximum possible final value of $\mathrm{LIS}(A) + \mathrm{LIS}(B)$.

Constraints

• $2\leq N\leq 3\times 10^5$
• $1\leq A_i\leq N$
• $1\leq B_i\leq N$
• $A_i\neq A_j$ and $B_i\neq B_j$, if $i\neq j$.

Input

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

$N$

$A_1$ $\ldots$ $A_N$

$B_1$ $\ldots$ $B_N$

Output

Print the maximum possible final value of $\mathrm{LIS}(A) + \mathrm{LIS}(B)$.

5
5 2 1 4 3
3 1 2 5 4

8

Here is one way to achieve $\mathrm{LIS}(A) + \mathrm{LIS}(B) = 8$.

• Do the operation with $i = 2$. Now, $A = (5,1,2,4,3)$, $B = (3,2,1,5,4)$.
• Do the operation with $i = 1$. Now, $A = (1,5,2,4,3)$, $B = (2,3,1,5,4)$.
• Do the operation with $i = 4$. Now, $A = (1,5,2,3,4)$, $B = (2,3,1,4,5)$.

Here, $A$ has a longest increasing subsequence $(1,2,3,4)$, so $\mathrm{LIS}(A)=4$, and $B$ has a longest increasing subsequence $(2,3,4,5)$, so $\mathrm{LIS}(B)=4$.

5
1 2 3 4 5
1 2 3 4 5

10

You can decide not to perform the operation at all to achieve $\mathrm{LIS}(A) + \mathrm{LIS}(B) = 10$.