CF988D Points and Powers of Two 数学结论题 规律 第十题

Points and Powers of Two

time limit per test

4 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

There are $\mathrm{nn distinct\; points\; on\; a\; coordinate\; line,\; the\; coordinate\; of ii-th\; point\; equals\; to xixi.\; Choose\; a\; subset\; of\; the\; given\; set\; of\; points\; such\; that\; the\; distance\; between\; each\; pair\; of\; points\; in\; a\; subset\; is\; an\; integral\; power\; of\; two.\; It\; is\; necessary\; to\; consider\; each\; pair\; of\; points,\; not\; only\; adjacent.\; Note\; that\; any\; subset\; containing\; one\; element\; satisfies\; the\; condition\; above.\; Among\; all\; these\; subsets,\; choose\; a\; subset\; with\; maximum\; possible\; size.}$

In other words, you have to choose the maximum possible number of points ${\mathrm{xi1,xi2,\dots ,ximxi1,xi2,\dots ,xim such\; that\; for\; each\; pair xijxij, xikxik it\; is\; true\; that |xij-xik|=2d|xij-xik|=2d where dd is\; some\; non-negative\; integer\; number\; (not\; necessarily\; the\; same\; for\; each\; pair\; of\; points).}}^{}$

Input

The first line contains one integer $\mathrm{nn (1\le n\le 2\cdot 1051\le n\le 2\cdot 105)\; —\; the\; number\; of\; points.}$

The second line contains $\mathrm{nn pairwise\; distinct\; integers x1,x2,\dots ,xnx1,x2,\dots ,xn (-109\le xi\le 109-109\le xi\le 109)\; —\; the\; coordinates\; of\; points.}$

Output

In the first line print $\mathrm{mm —\; the\; maximum\; possible\; number\; of\; points\; in\; a\; subset\; that\; satisfies\; the\; conditions\; described\; above.}$

In the second line print $\mathrm{mm integers\; —\; the\; coordinates\; of\; points\; in\; the\; subset\; you\; have\; chosen.}$

If there are multiple answers, print any of them.

Examples

input

Copy

6
3 5 4 7 10 12

output

Copy

3
7 3 5

input

Copy

5
-1 2 5 8 11

output

Copy

1
8

Note

In the first example the answer is $[7,3,5][7,3,5].\; Note,\; that |7-3|=4=22|7-3|=4=22, |7-5|=2=21|7-5|=2=21 and |3-5|=2=21|3-5|=2=21.\; You\; can\text{'}t\; find\; a\; subset\; having\; more\; points\; satisfying\; the\; required\; property.$