English · Original
Chinese · Translation
Formal · Original
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_.
For example, the prairie partitions of 12, 17, 7 and 1 are:
<center>12 = 1 + 2 + 4 + 5,17 = 1 + 2 + 4 + 8 + 2,
7 = 1 + 2 + 4,
1 = 1.
</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!
## Input
The first line contains a single integer _n_ (1 ≤ _n_ ≤ 105) — the number of numbers given from Alice to Borys.
The 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.
## Output
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.
If there are no such values of _m_, output a single integer _\-1_.
[samples]
## Note
In the first example, Alice could get the input sequence from \[6, 20\] as the original sequence.
In the second example, Alice's original sequence could be either \[4, 5\] or \[3, 3, 3\].
可以证明，任何一个正整数 $x$ 都可以唯一地表示为 $x = 1 + 2 + 4 + \dots + 2^{k-1} + r$，其中 $k$ 和 $r$ 是整数，满足 $k \geq 0$ 且 $0 < r \leq 2^k$。我们将这种表示称为 $x$ 的 _prairie partition_。
例如，12、17、7 和 1 的 prairie partition 分别为：
$17 = 1 + 2 + 4 + 8 + 2$，
$7 = 1 + 2 + 4$，
$1 = 1$。
Alice 取了一个正整数序列（可能包含重复元素），将每个元素替换为其 prairie partition 中的加数，然后将得到的所有数字按非降序排列，并将这个序列交给了 Borys。现在 Borys 想知道，Alice 原始序列可能包含多少个元素？请找出所有可能的取值！
第一行包含一个整数 $n$（$1 \leq n \leq 10^5$）——Alice 交给 Borys 的数字个数。
第二行包含 $n$ 个整数 $a_1, a_2, \dots, a_n$（$1 \leq a_i \leq 10^{12}$；$a_1 \leq a_2 \leq \dots \leq a_n$）——Alice 交给 Borys 的数字。
请按**递增顺序**输出所有可能的 $m$ 值，使得存在一个长度为 $m$ 的正整数序列，当将每个元素替换为其 prairie partition 的加数，并将结果按非降序排列后，恰好得到输入的序列。
如果没有这样的 $m$ 值，请输出单个整数 _-1_。
在第一个例子中，Alice 可能的原始序列为 $[6, 20]$。
在第二个例子中，Alice 的原始序列可能是 $[4, 5]$ 或 $[3, 3, 3]$。
## Input
第一行包含一个整数 $n$（$1 \leq n \leq 10^5$）——Alice 交给 Borys 的数字个数。第二行包含 $n$ 个整数 $a_1, a_2, \dots, a_n$（$1 \leq a_i \leq 10^{12}$；$a_1 \leq a_2 \leq \dots \leq a_n$）——Alice 交给 Borys 的数字。
## Output
请按**递增顺序**输出所有可能的 $m$ 值，使得存在一个长度为 $m$ 的正整数序列，当将每个元素替换为其 prairie partition 的加数，并将结果按非降序排列后，恰好得到输入的序列。如果没有这样的 $m$ 值，请输出单个整数 _-1_。
[samples]
## Note
在第一个例子中，Alice 可能的原始序列为 $[6, 20]$。在第二个例子中，Alice 的原始序列可能是 $[4, 5]$ 或 $[3, 3, 3]$。
**Definitions**
Let $ \mathcal{P}(x) $ denote the *prairie partition* of a positive integer $ x $, uniquely defined as:
$$
x = 1 + 2 + 4 + \dots + 2^{k-1} + r = (2^k - 1) + r, \quad \text{where } k \geq 0, \; 0 < r \leq 2^k.
$$
The multiset of summands in $ \mathcal{P}(x) $ is $ \{1, 2, 4, \dots, 2^{k-1}, r\} $.
Let $ A = (a_1, a_2, \dots, a_n) $ be the input sequence, sorted non-decreasingly, representing the multiset of all summands from the prairie partitions of some original sequence $ B = (b_1, \dots, b_m) $ of positive integers.
**Constraints**
1. $ 1 \leq n \leq 10^5 $
2. $ 1 \leq a_i \leq 10^{12} $ for all $ i $
3. $ a_1 \leq a_2 \leq \dots \leq a_n $
**Objective**
Find all possible values of $ m \in \mathbb{Z}^+ $ such that there exists a sequence $ B = (b_1, \dots, b_m) $ of positive integers where the multiset union of $ \mathcal{P}(b_1) \cup \mathcal{P}(b_2) \cup \dots \cup \mathcal{P}(b_m) $, when sorted, equals $ A $.
Output all such $ m $ in increasing order; if none exist, output $-1$.
API Response (JSON)
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"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 representa...",
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"content": "可以证明，任何一个正整数 $x$ 都可以唯一地表示为 $x = 1 + 2 + 4 + \\dots + 2^{k-1} + r$，其中 $k$ 和 $r$ 是整数，满足 $k \\geq 0$ 且 $0 < r \\leq 2^k$。我们将这种表示称为 $x$ 的 _prairie partition_。\n\n例如，12、17、7 和 1 的 prairie partition 分别为：\n\n$17 = ...",
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"content": "**Definitions** \nLet $ \\mathcal{P}(x) $ denote the *prairie partition* of a positive integer $ x $, uniquely defined as: \n$$\nx = 1 + 2 + 4 + \\dots + 2^{k-1} + r = (2^k - 1) + r, \\quad \\text{where } ...",
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