CF2195D Absolute Cinema

There is a hidden sequence $ a_1,a_2,\ldots,a_n $ of $ n $ integers ( $ n \ge 2 $ ). It is guaranteed that $ |a_i| \le 1000 $ for all $ 1 \le i \le n $ . Let's define a function $ f(x) $ as follows: $$$ f(x)=\sum_{i=1}^n a_i \cdot |i-x| $$$ Given $ n $ values $ f(1),f(2),\ldots,f(n) $ , please determine the values of $ a_1,a_2,\ldots,a_n $ . It is guaranteed that the values $ a_1,a_2,\ldots,a_n $ can be determined uniquely.

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 a single integer $ n $ ( $ \color{red}{2} \le n \le 300\,000 $ ). The second line of each test case contains $ n $ integers $ f(1),f(2),\ldots,f(n) $ ( $ -10^{14} \le f(i) \le 10^{14} $ ). It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 300\,000 $ .

For each test case, output $ n $ integers $ a_1,a_2,\ldots,a_n $ on a separate line ( $ |a_i| \le 1000 $ ). It is guaranteed that the values $ a_1,a_2,\ldots,a_n $ can be determined uniquely.

In the first test case, the hidden sequence is $ a=[1,4,2,3] $ . The values $ f(1),f(2),\ldots,f(n) $ are as follows: - $ f(1) = 1 \cdot |1-1| + 4 \cdot |2-1| + 2 \cdot |3-1| + 3 \cdot |4-1| = 0+4+4+9 = 17 $ ; - $ f(2) = 1 \cdot |1-2| + 4 \cdot |2-2| + 2 \cdot |3-2| + 3 \cdot |4-2| = 1+0+2+6 = 9 $ ; - $ f(3) = 1 \cdot |1-3| + 4 \cdot |2-3| + 2 \cdot |3-3| + 3 \cdot |4-3| = 2+4+0+3 = 9 $ ; - $ f(4) = 1 \cdot |1-4| + 4 \cdot |2-4| + 2 \cdot |3-4| + 3 \cdot |4-4| = 3+8+2+0 = 13 $ .