{"raw_statement":[{"iden":"statement","content":"It can be shown that any positive integer _x_ can be uniquely represented as _x_ = 1 + 2 + 4 + ... + 2_k_ - 1 + _r_, where _k_ and _r_ are integers, _k_ ≥ 0, 0 < _r_ ≤ 2_k_. Let's call that representation prairie partition of _x_.\n\nFor example, the prairie partitions of 12, 17, 7 and 1 are:\n\n<center>12 = 1 + 2 + 4 + 5,17 = 1 + 2 + 4 + 8 + 2,\n\n7 = 1 + 2 + 4,\n\n1 = 1.\n\n</center>Alice took a sequence of positive integers (possibly with repeating elements), replaced every element with the sequence of summands in its prairie partition, arranged the resulting numbers in non-decreasing order and gave them to Borys. Now Borys wonders how many elements Alice's original sequence could contain. Find all possible options!"},{"iden":"input","content":"The first line contains a single integer _n_ (1 ≤ _n_ ≤ 105) — the number of numbers given from Alice to Borys.\n\nThe second line contains _n_ integers _a_1, _a_2, ..., _a__n_ (1 ≤ _a__i_ ≤ 1012; _a_1 ≤ _a_2 ≤ ... ≤ _a__n_) — the numbers given from Alice to Borys."},{"iden":"output","content":"Output, **in increasing order**, all possible values of _m_ such that there exists a sequence of positive integers of length _m_ such that if you replace every element with the summands in its prairie partition and arrange the resulting numbers in non-decreasing order, you will get the sequence given in the input.\n\nIf there are no such values of _m_, output a single integer _\\-1_."},{"iden":"examples","content":"Input\n\n8\n1 1 2 2 3 4 5 8\n\nOutput\n\n2 \n\nInput\n\n6\n1 1 1 2 2 2\n\nOutput\n\n2 3 \n\nInput\n\n5\n1 2 4 4 4\n\nOutput\n\n\\-1"},{"iden":"note","content":"In the first example, Alice could get the input sequence from \\[6, 20\\] as the original sequence.\n\nIn the second example, Alice's original sequence could be either \\[4, 5\\] or \\[3, 3, 3\\]."}],"translated_statement":null,"sample_group":[],"show_order":[],"formal_statement":null,"simple_statement":null,"has_page_source":false}