It can be shown that any positive integer *x* can be uniquely represented as *x* = 1 + 2 + 4 + ... + 2<sup class="upper-index">*k* - 1</sup> + *r*, where *k* and *r* are integers, *k* ≥ 0, 0 &lt; *r* ≤ 2<sup class="upper-index">*k*</sup>. Let's call that representation prairie partition of *x*.

For example, the prairie partitions of 12, 17, 7 and 1 are: 

17 = 1 + 2 + 4 + 8 + 2,

7 = 1 + 2 + 4,

1 = 1. 

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!
The first line contains a single integer *n* (1 ≤ *n* ≤ 10<sup class="upper-index">5</sup>) — the number of numbers given from Alice to Borys.

The second line contains *n* integers *a*<sub class="lower-index">1</sub>, *a*<sub class="lower-index">2</sub>, ..., *a*<sub class="lower-index">*n*</sub> (1 ≤ *a*<sub class="lower-index">*i*</sub> ≤ 10<sup class="upper-index">12</sup>; *a*<sub class="lower-index">1</sub> ≤ *a*<sub class="lower-index">2</sub> ≤ ... ≤ *a*<sub class="lower-index">*n*</sub>) — the numbers given from Alice to Borys.
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].
8
1 1 2 2 3 4 5 8
2 
6
1 1 1 2 2 2
2 3 
5
1 2 4 4 4
-1
20
1 1 1 1 2 2 2 2 4 4 4 4 8 8 8 8 8 10 10 11
4 
20
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2
9 10 11 12 13 14 15 16 17 
25
1 1 1 1 2 2 2 2 4 4 4 4 8 8 8 8 13 15 16 16 31 32 36 41 55
-1
25
1 1 1 1 2 2 2 2 4 4 4 4 8 8 8 9 16 16 32 40 43 53 61 64 128
-1
45
1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 8 8 8 8 8 8 16 16 16 16 16 32 32 32 32 32 41 64 64 64 64 128 128 128 256 256 512
5 
100
1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 6 8 8 8 8 8 8 8 10 16 16 16 16 16 16 16 17 22 24 24 30 32 32 32 32 32 32 48 62 64 64 65 65 67 70 74 88 89 98 99 101 101 109 121 127 128 128 137 143 152 153 155 156 160 161 170 183 186 196 196 214 220 226 228 230 238 240 241 245 249 249 250 253 254 256 256 512 1024 1703
-1
101
1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 7 8 8 8 8 8 8 8 8 8 8 8 16 16 16 16 16 16 16 16 16 16 16 32 32 32 32 32 32 32 32 32 32 64 64 64 64 64 64 64 64 64 128 128 128 128 128 128 128 128 239 256 256 256 256 256 373 512 512 512 512 695 1024 1024 1024
11 12 
101
1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 8 8 8 8 8 8 8 8 8 8 8 8 16 16 16 16 16 16 16 16 16 16 16 16 26 32 32 32 32 32 32 32 32 49 58 64 64 64 64 64 64 122 128 128 128 128 128 128 256 256 256 256 256 491 512 512 512 512 718 1024 1024 1024 2935 3123
-1
100
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 8 8 8 8 8 8 8 8 11 13 14
27 28 29 30 31 32 33 34 35 36 37 38 
1
1
1 
2
1 2
1 
2
1 1
1 2 
1
2
-1
1
1000000000000
-1
2
1 1000000000000
-1
40
1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216 33554432 67108864 134217728 268435456 536870912 1073741824 2147483648 4294967296 8589934592 17179869184 34359738368 68719476736 137438953472 274877906944 549755813888
1 
41
1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216 33554432 67108864 134217728 268435456 536870912 1073741824 2147483648 4294967296 8589934592 17179869184 34359738368 68719476736 137438953472 274877906944 549755813888 1000000000000
1 
41
1 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216 33554432 67108864 134217728 268435456 536870912 1073741824 2147483648 4294967296 8589934592 17179869184 34359738368 68719476736 137438953472 274877906944 549755813888
1 2 
96
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 8 9 12 12 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 18 21
-1
print("_RANDOM_GUESS_1689343188.0738711")# 1689343188.07391