CF1536A Omkar and Bad Story

Omkar has received a message from Anton saying "Your story for problem A is confusing. Just make a formal statement." Because of this, Omkar gives you an array $ a = [a_1, a_2, \ldots, a_n] $ of $ n $ distinct integers. An array $ b = [b_1, b_2, \ldots, b_k] $ is called nice if for any two distinct elements $ b_i, b_j $ of $ b $ , $ |b_i-b_j| $ appears in $ b $ at least once. In addition, all elements in $ b $ must be distinct. Can you add several (maybe, $ 0 $ ) integers to $ a $ to create a nice array $ b $ of size at most $ 300 $ ? If $ a $ is already nice, you don't have to add any elements. For example, array $ [3, 6, 9] $ is nice, as $ |6-3|=|9-6| = 3 $ , which appears in the array, and $ |9-3| = 6 $ , which appears in the array, while array $ [4, 2, 0, 6, 9] $ is not nice, as $ |9-4| = 5 $ is not present in the array. For integers $ x $ and $ y $ , $ |x-y| = x-y $ if $ x > y $ and $ |x-y| = y-x $ otherwise.

Each test contains multiple test cases. The first line contains $ t $ ( $ 1 \leq t \leq 50 $ ), the number of test cases. Description of the test cases follows. The first line of each test case contains a single integer $ n $ ( $ 2 \leq n \leq 100 $ ) — the length of the array $ a $ . The second line of each test case contains $ n $ distinct integers $ a_1, a_2, \cdots, a_n $ ( $ -100 \leq a_i \leq 100 $ ) — the elements of the array $ a $ .

For each test case, output one line containing YES if Omkar can create a nice array $ b $ by adding elements to $ a $ and NO otherwise. The case of each letter does not matter, so yEs and nO will also be accepted. If the first line is YES, output a second line containing a single integer $ k $ ( $ n \leq k \leq 300 $ ). Then output one line containing $ k $ distinct integers $ b_1, b_2, \cdots, b_k $ ( $ -10^9 \leq b_i \leq 10^9 $ ), the elements of the nice array $ b $ . $ b_1, b_2, \cdots, b_k $ can be in any order. For each $ a_i $ in $ a $ , $ a_i $ must appear at least once in $ b $ . It can be proved that if Omkar can create such an array $ b $ , then he can also do so in a way that satisfies the above constraints. If multiple solutions exist, you can print any.

For the first case, you can add integers to $ a $ to receive the array $ b = [6, 0, 3, 9] $ . Note that $ |6-3| = |9-6| = |3-0| = 3 $ and $ 3 $ is in $ b $ , $ |6-0| = |9-3| = 6 $ and $ 6 $ is in $ b $ , and $ |9-0| = 9 $ is in $ b $ , so $ b $ is nice. For the second case, you can add integers to $ a $ to receive the array $ b = [5, 3, 1, 2, 4] $ . We have that $ |2-1| = |3-2| = |4-3| = |5-4| = 1 $ is in $ b $ , $ |3-1| = |4-2| = |5-3| = 2 $ is in $ b $ , $ |4-1| = |5-2| = 3 $ is in $ b $ , and $ |5-1| = 4 $ is in $ b $ , so $ b $ is nice. For the fourth case, you can add integers to $ a $ to receive the array $ b = [8, 12, 6, 2, 4, 10] $ . We have that $ |4-2| = |6-4| = |8-6| = |10-8| = |12-10| = 2 $ is in $ b $ , $ |6-2| = |8-4| = |10-6| = |12-8| = 4 $ is in $ b $ , $ |8-2| = |10-4| = |12-6| = 6 $ is in $ b $ , $ |10-2| = |12-4| = 8 $ is in $ b $ , and $ |12-2| = 10 $ is in $ b $ , so $ b $ is nice. It can be proven that for all other test cases it is impossible to create a nice array $ b $ .