Double Sort II
You are given two permutations $$$a$$$ and $$$b$$$, both of size $$$n$$$. A permutation of size $$$n$$$ is an array of $$$n$$$ elements, where each integer from $$$1$$$ to $$$n$$$ appears exactly once. The elements in each permutation are indexed from $$$1$$$ to $$$n$$$.You can perform the following operation any number of times:  choose an integer $$$i$$$ from $$$1$$$ to $$$n$$$;  let $$$x$$$ be the integer such that $$$a_x = i$$$. Swap $$$a_i$$$ with $$$a_x$$$;  let $$$y$$$ be the integer such that $$$b_y = i$$$. Swap $$$b_i$$$ with $$$b_y$$$. Your goal is to make both permutations sorted in ascending order (i. e. the conditions $$$a_1 < a_2 < \dots < a_n$$$ and $$$b_1 < b_2 < \dots < b_n$$$ must be satisfied) using minimum number of operations. Note that both permutations must be sorted after you perform the sequence of operations you have chosen.
The first line contains one integer $$$n$$$ ($$$2 \le n \le 3000$$$).The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le n$$$; all $$$a_i$$$ are distinct).The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le n$$$; all $$$b_i$$$ are distinct).
First, print one integer $$$k$$$ ($$$0 \le k \le 2n$$$) — the minimum number of operations required to sort both permutations. Note that it can be shown that $$$2n$$$ operations are always enough.Then, print $$$k$$$ integers $$$op_1, op_2, \dots, op_k$$$ ($$$1 \le op_j \le n$$$), where $$$op_j$$$ is the value of $$$i$$$ you choose during the $$$j$$$-th operation.If there are multiple answers, print any of them.
import sys, os, io
input = io.BytesIO(os.read(0, os.fstat(0).st_size)).readline

def get_root(s):
    v = []
    while not s == root[s]:
        v.append(s)
        s = root[s]
    for i in v:
        root[i] = s
    return s

def unite(s, t):
    (rs, rt) = (get_root(s), get_root(t))
    if not rs ^ rt:
        return
    if rank[s] == rank[t]:
        rank[rs] += 1
    if rank[s] >= rank[t]:
        root[rt] = rs
        size[rs] += size[rt]
    else:
        root[rs] = rt
        size[rt] += size[rs]
    return

def same(s, t):
    return True if get_root(s) == get_root(t) else False

def get_size(s):
    return size[get_root(s)]

def add_edge(u, v, c):
    G[u].append([v, c, len(G[v]), 1])
    G[v].append([u, 0, len(G[u]) - 1, 0])
    return

def bfs(s):
    (q, k) = ([s], 0)
    dist = [inf] * l
    dist[s] = 0
    while len(q) ^ k:
        i = q[k]
        di = dist[i]
        for (j, c, _, _) in G[i]:
            if not dist[j] ^ inf and c:
                dist[j] = di + 1
                q.append(j)
        k += 1
    return dist

def dfs(dist, s, t, nowe):
    mi = [inf] * l
    st = [s]
    se = set()
    se.add(s)
    d = []
    while st:
        u = st[-1]
        if u == t:
            st.pop()
            break
        if nowe[u] < len(G[u]):
            (v, c, _, _) = G[u][nowe[u]]
            if c and (not dist[v] ^ dist[u] + 1) and (not v in se):
                st.append(v)
                d.append(nowe[u])
                se.add(v)
                mi[v] = min(mi[u], c)
            else:
                nowe[u] += 1
        else:
            se.remove(st.pop())
            if d:
                d.pop()
            if st:
                nowe[st[-1]] += 1
    return (st, d, mi[t])

def max_flow(s, t):
    ans = 0
    while True:
        dist = bfs(s)
        if not dist[t] ^ inf:
            break
        nowe = [0] * l
        while True:
            (st, d, f) = dfs(dist, s, t, nowe)
            if not f % inf:
                break
            ans += f
            for (u, i) in zip(st, d):
                G[u][i][1] -= f
                G[G[u][i][0]][G[u][i][2]][1] += f
    return ans
n = int(input())
a = [0] + list(map(int, input().split()))
b = [0] + list(map(int, input().split()))
root = [i for i in range(n + 1)]
rank = [1 for _ in range(n + 1)]
size = [1 for _ in range(n + 1)]
for i in range(1, n + 1):
    unite(i, a[i])
u = []
for i in range(1, n + 1):
    u.append(get_root(i))
s = list(set(u))
d = dict()
lu = len(s)
for i in range(lu):
    d[s[i]] = i
for i in range(n):
    u[i] = d[u[i]]
root = [i for i in range(n + 1)]
rank = [1 for _ in range(n + 1)]
size = [1 for _ in range(n + 1)]
for i in range(n + 1):
    unite(i, b[i])
v = []
for i in range(1, n + 1):
    v.append(get_root(i))
s = list(set(v))
d = dict()
lv = len(s)
for i in range(lv):
    d[s[i]] = i
for i in range(n):
    v[i] = d[v[i]]
d = dict()
inf = pow(10, 9) + 1
l = lu + lv + 2
(s, t) = (l - 1, l - 2)
G = [[] for _ in range(l)]
for i in range(n):
    (ui, vi) = (u[i], v[i])
    if (ui, vi) in d:
        continue
    d[ui, vi] = i + 1
    add_edge(ui, vi + lu, 1)
for i in range(lu):
    add_edge(s, i, 1)
for i in range(lv):
    add_edge(i + lu, t, 1)
f = max_flow(s, t)
ng = [0] * (n + 1)
for i in range(lu):
    for (j, c, _, _) in G[i]:
        if j ^ s and (not c):
            ng[d[i, j - lu]] = 1
            break
ans = []
for i in range(1, n + 1):
    if not ng[i]:
        ans.append(i)
k = len(ans)
print(k)
sys.stdout.write(' '.join(map(str, ans)))
import sys
import math
from collections import deque
input = lambda : sys.stdin.readline().strip()
mod = 998244353
t = 1

def match(x):
    for y in G[x]:
        if used[y[0]] == 0:
            used[y[0]] = 1
            if right[y[0]] == 0 or match(right[y[0]][0]):
                right[y[0]] = (x, y[1])
                return True
    return False
for _ in range(t):
    n = int(input())
    alist = [0] + list(map(int, input().split()))
    blist = [0] + list(map(int, input().split()))
    now = 0
    avisited = [0] * (n + 1)
    acir = [-1] * (n + 1)
    for i in range(1, n + 1):
        if avisited[i]:
            continue
        avisited[i] = 1
        acir[i] = now
        while not avisited[alist[i]]:
            i = alist[i]
            avisited[i] = 1
            acir[i] = now
        now += 1
    G = [[] for i in range(n)]
    bvisited = [0] * (n + 1)
    anow = 0
    for i in range(1, n + 1):
        if bvisited[i]:
            continue
        bvisited[i] = 1
        G[acir[i]].append((anow, i))
        while not bvisited[blist[i]]:
            i = blist[i]
            bvisited[i] = 1
            G[acir[i]].append((anow, i))
        anow += 1
    right = [0] * n
    temp = 0
    for i in range(n):
        used = [0] * n
        if match(i):
            temp += 1
    print(n - temp)
    sign = [1] * (n + 1)
    for i in range(n):
        if right[i]:
            sign[right[i][1]] = 0
    ans = []
    for i in range(1, n + 1):
        if sign[i]:
            ans.append(i)
    print(*ans)
import sys
input = lambda : sys.stdin.readline().rstrip()
N = int(input())
A = [int(x) - 1 for x in input().split()]
B = [int(x) - 1 for x in input().split()]

def find_cycle(arr):
    sz = 0
    comp = [-1] * N
    cycles = []
    for (i, a) in enumerate(arr):
        if comp[i] == -1:
            curr = [i]
            comp[i] = sz
            j = a
            while comp[j] == -1:
                curr.append(j)
                comp[j] = sz
                j = arr[j]
            cycles.append(curr)
            sz += 1
    return [comp, cycles]
(X, cx) = find_cycle(A)
(Y, cy) = find_cycle(B)
n = len(cx)
k = len(cy)
G = [[] for _ in range(n)]
for i in range(N):
    G[X[i]].append(Y[i])
mt = [-1] * k

def khun(v):
    if used[v]:
        return False
    used[v] = True
    for u in G[v]:
        if mt[u] == -1 or khun(mt[u]):
            mt[u] = v
            return True
    return False
for v in range(n):
    used = [False] * n
    khun(v)
match = [False] * N
for i in range(k):
    if mt[i] != -1:
        for digit in cx[mt[i]]:
            if digit in cy[i]:
                match[digit] = True
                break
res = [i + 1 for i in range(N) if not match[i]]
print(len(res))
print(' '.join(map(str, res)))