CF2258E DivMEX

You are given an array $ a_1, a_2, \ldots, a_n $ of positive integers. Let $ f(l,r) $ for $ 1 \leq l \leq r \leq n $ be the smallest positive integer that doesn't divide $ \operatorname{lcm}(a_l, a_{l+1}, \ldots, a_r) $ . Here, $ \operatorname{lcm} $ of an array of integers denotes the [least common multiple (LCM)](https://en.wikipedia.org/wiki/Least_common_multiple) of these integers. Determine every positive integer $ x $ such that there exists a pair of integers $ (l,r) $ such that $ 1 \leq l \leq r \leq n $ and $ f(l,r) = x $ .

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10^4 $ ). The description of the test cases follows. The first line of each test case contains $ n $ ( $ 1 \le n \le 2 \cdot 10^5 $ ), denoting the size of the array. The second line of each test case contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 1 \le a_i \le n $ ). It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case, output two lines. The first line should contain a single integer $ k $ — the number of suitable integers $ x $ . The second line should contain $ k $ integers $ x_1 \lt x_2 \lt \ldots \lt x_k $ , the suitable integers in increasing order.

In the first test case, the only $ (l,r) $ pair is $ (1,1) $ , and $ f(1,1) = 2 $ . In the second test case, $ (l,r)=(1,1) $ yields the set of divisors $ \{ 1,2 \} $ , $ (l,r)=(1,2) $ yields the set of divisors $ \{ 1, 2, 3, 6\} $ , and $ (l,r)=(2,2) $ yields the set of divisors $ \{ 1, 3 \} $ . This means $ f(1,1)=3 $ , $ f(1,2)=4 $ , and $ f(2,2)=2 $ . Since no other $ (l,r) $ pair produces a different $ f $ value, the answer is $ [2,3,4] $ . For the last test case, $ f(2,2)=2 $ , $ f(1,1)=3 $ , and $ f(1,5)=5 $ . It can be seen that no other values can be obtained.