{"problem":{"name":"D. Regular Bracket Sequence Again?","description":{"content":"You have a collection of $N$ opening brackets '_(_' and $N$ closing brackets '_)_'. A bracket sequence is $N$ periodic if it has a period of length $N$. Example - \"_)()(_\" $N = 2$. It is $N$ periodi","description_type":"Markdown"},"platform":"Codeforces","limit":{"time_limit":1000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"CF10276D"},"statements":[{"statement_type":"Markdown","content":"You have a collection of $N$ opening brackets '_(_' and $N$ closing brackets '_)_'.\n\nA bracket sequence is $N$ periodic if it has a period of length $N$.\n\nExample - \"_)()(_\" $N = 2$. It is $N$ periodic but \"_)(()_\" is not.\n\nYour task is to find out how many distinct *regular bracket sequence* can be formed using these $2 N$ brackets such that sequence is *not $N$ periodic*.\n\nAs the answer can be rather large, print the remainder after dividing it by $1000000007 (10^9 + 7)$.\n\nRecall what the regular bracket sequence is:\n\nFor example, \"_()()_\", \"_(())()_\", \"_(())_\" and \"_()_\" are regular bracket sequences, but \"_)(_\", \"_()(_\" and \"_)))_\" are not.\n\nThe first line contains a single integer $T (1 <= T <= 10^3)$ — the number of test cases in the input. Then $T$ test cases follow.\n\nEach query contains one integer $N (1 <= N <= 10^3)$: the number of opening and closing brackets you have.\n\nFor each test from the input print the number of distinct regular bracket sequence we can form using $N$ opening bracket and $N$ closing bracket and sequence is not $N$ periodic modulo $1000000007 (10^9 + 7)$. \n\nPrint the answers to the tests in the order in which the tests are given in the input.\n\nIn the first query — The regular bracket sequence which can be formed by $1$ opening bracket and $1$ closing bracket is \"_()_\".\n\nIn the second query — The regular bracket sequence which can be formed by $2$ opening bracket and $2$ closing bracket are \"_(())_\" and \"_()()_\". We discard second one because \"_()()_\" is $2$ periodic.\n\nIn the third query — The regular bracket sequence which can be formed by $3$ opening bracket and $3$ closing bracket are \"_((()))_\" , \"_(())()_\" , \"_()(())_\" , \"_()()()_\" and \"_(()())_\". No one is $3$ periodic.\n\n## Input\n\nThe first line contains a single integer $T (1 <= T <= 10^3)$ — the number of test cases in the input. Then $T$ test cases follow.Each query contains one integer $N (1 <= N <= 10^3)$: the number of opening and closing brackets you have.\n\n## Output\n\nFor each test from the input print the number of distinct regular bracket sequence we can form using $N$ opening bracket and $N$ closing bracket and sequence is not $N$ periodic modulo $1000000007 (10^9 + 7)$. Print the answers to the tests in the order in which the tests are given in the input.\n\n[samples]\n\n## Note\n\nIn the first query — The regular bracket sequence which can be formed by $1$ opening bracket and $1$ closing bracket is \"_()_\".In the second query — The regular bracket sequence which can be formed by $2$ opening bracket and $2$ closing bracket are \"_(())_\" and \"_()()_\". We discard second one because \"_()()_\" is $2$ periodic.In the third query — The regular bracket sequence which can be formed by $3$ opening bracket and $3$ closing bracket are \"_((()))_\" , \"_(())()_\" , \"_()(())_\" , \"_()()()_\" and \"_(()())_\". No one is $3$ periodic.","is_translate":false,"language":"English"},{"statement_type":"Markdown","content":"**Definitions**  \nLet $ T \\in \\mathbb{Z} $ be the number of test cases.  \nFor each test case:  \n- Let $ n \\in \\mathbb{Z} $ be the number of defensive towers.  \n- Let $ P = \\{(x_i, y_i, d_i) \\mid i \\in \\{1, \\dots, n\\}\\} $ be the set of towers, where:  \n  - $ (x_i, y_i) \\in \\{1, \\dots, n\\}^2 $ is the position of tower $ i $,  \n  - $ d_i \\in \\{1, 2, 3, 4\\} $ is its direction, mapping to:  \n    - $ d_i = 1 $: protects $ [x_i, n+1] \\times [y_i, n+1] $ (northeast),  \n    - $ d_i = 2 $: protects $ [0, x_i] \\times [y_i, n+1] $ (northwest),  \n    - $ d_i = 3 $: protects $ [0, x_i] \\times [0, y_i] $ (southwest),  \n    - $ d_i = 4 $: protects $ [x_i, n+1] \\times [0, y_i] $ (southeast).  \n\n**Constraints**  \n1. $ 1 \\le T \\le 10^4 $  \n2. $ 1 \\le n \\le 10^6 $ per test case  \n3. $ 1 \\le x_i, y_i \\le n $  \n4. $ 1 \\le d_i \\le 4 $  \n5. $ \\sum_{\\text{all test cases}} n \\le 5 \\times 10^6 $  \n\n**Objective**  \nFind the minimum number of towers to cover the entire rectangular region $ [0, n+1] \\times [0, n+1] $.  \nIf impossible, output \"Impossible\".","is_translate":false,"language":"Formal"}],"meta":{"iden":"CF10276D","tags":[],"sample_group":[],"created_at":"2026-03-03 11:00:39"}}