Problem: There are n distinct points on a coordinate line, the coordinate of i-th point equals to x_i. 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 x_{i_1}, x_{i_2}, ..., x_{i_m} such that for each pair x_{i_j}, x_{i_k} it is true that |x_{i_j} - x_{i_k}| = 2^d where d is some non-negative integer number (not necessarily the same for each pair of points).

Input

The first line contains one integer n (1 ≤ n ≤ 2 ⋅ 10^5) — the number of points.

The second line contains n pairwise distinct integers x_1, x_2, ..., x_n (-10^9 ≤ x_i ≤ 10^9) — the coordinates of points.

Output

In the first line print m — the maximum possible number of points in a subset that satisfies the conditions described above.

In the second line print m integers — the coordinates of points in the subset you have chosen.

If there are multiple answers, print any of them.

Examples

Input

6
3 5 4 7 10 12


Output

3
7 3 5

Input

5
-1 2 5 8 11


Output

1
8

Note

In the first example the answer is [7, 3, 5]. Note, that |7-3|=4=2^2, |7-5|=2=2^1 and |3-5|=2=2^1. You can't find a subset having more points satisfying the required property.

Solution (in JAVA):

Solution:
import java.io.*;
import java.math.BigInteger;
import java.util.*;



public class Main {

	public static void main(String[] args) throws IOException {
		InputStream inputStream = System.in;
		OutputStream outputStream = System.out;
		InputReader in = new InputReader(inputStream);
		PrintWriter out = new PrintWriter(outputStream);
		Task t = new Task();
		t.solve(in, out);
		out.close();
	}
	
	static class ListNode {
		 int val;
		 ListNode next;
		 ListNode(int x) { val = x; }
	}		

	static class Task {
		public void solve(InputReader in, PrintWriter out) throws IOException {
			long tStart = System.currentTimeMillis();
			//System.out.println((long)Math.pow(2, 31));
			int n = in.nextInt();
			int arr[] = new int[n];
			HashMap<Integer,Integer> mp = new HashMap<Integer,Integer>();
			for(int i=0;i<n;i++){
				arr[i] = in.nextInt();
				mp.put(arr[i], 1);
			}
			int re[] = new int[3];
			re[0] = arr[0];
			int x=1;
			for(int i=0;i<n;i++){
				for(int j=0;j<=31;j++){
					int gap = (int) Math.pow(2, j);
					if(x<2){
						if(mp.containsKey(arr[i]+gap)){
							x=2;
							re[0] = arr[i];
							re[1] = arr[i]+gap;
						}else if(mp.containsKey(arr[i]-gap)){
							x=2;
							re[0] = arr[i];
							re[1] = arr[i]-gap;
						}
					}
					if(x<3){
						if(mp.containsKey(arr[i]+gap)&&mp.containsKey(arr[i]-gap)){
							x=3;
							re[0] = arr[i];
							re[1] = arr[i]-gap;
							re[2] = arr[i]+gap;
							out.println(x);
							for(int p=0;p<x;p++){
								out.print(re[p]+" ");
							}
							out.println();		
							return;
						}						
					}
				}
			}
			out.println(x);
			for(int i=0;i<x;i++){
				out.print(re[i]+" ");
			}
			out.println();
			//Dumper.print(shortestPathLength(arr));
		}
	
	    public int shortestPathLength(int[][] graph) {
	        int n = graph.length;
	        if(n==1) return 0;
	        int g[][] = new int[n][n];
	        for(int i=0;i<n;i++){
	        	Arrays.fill(g[i], 99999999);
	        	g[i][i] = 0;
	        	for(int j:graph[i]){
	        		g[i][j] = 1;
	        	}
	        }	        
	        for(int k=0;k<n;k++){
	        	for(int i=0;i<n;i++){
	        		for(int j=0;j<n;j++){
	        			g[i][j] = Math.min(g[i][j], g[i][k]+g[k][j]);
	        		}
	        	}
	        }
	        int p[] = new int[(int)Math.pow(2, n)];
	        Arrays.fill(p, 99999999);
	        int target = (int)Math.pow(2, n)-1;
	        for(int i=0;i<n;i++){
	        	//Dumper.print(Integer.toBinaryString(1<<i));
	        	dp(g,p,1<<i,i,0,target);
	        }      
	        
	        //Dumper.print_2d_arr(g, n, n);
	        Dumper.print_int_arr(p);
	        return p[target];
	    }		
	    int dp(int[][] g, int[] p, int status, int cur, int step, int target){
	    	if(p[status]<=step) return p[status];
	    	if(status==target) {
	    		Dumper.print(step);
	    		return p[status] = step;
	    	}
	    	int min = 99999999;
	    	for(int i=0;i<g.length;i++){
	    		if(g[cur][i]==0||(status & (1L << i)) != 0) continue;
	    		min = Math.min(min, dp(g,p,status|(1<<i),i,step+1,target));
	    	}
	    	return p[status] = min;
	    }
			
	    public double findMedianSortedArrays(int[] nums1, int[] nums2) {
	    	int l1=0; int l2=0; int r1=nums1.length-1; int r2=nums2.length-1;
	    	int n = nums1.length; int m = nums2.length;
	    	if(n==0){
	    		if(m==0) return 0.0;
	    		if(m%2==1) return nums2[m/2];
	    		else return 1.0*(nums2[m/2]+nums2[m/2-1])/2.0;
	    	}
	    	if(m==0){
	    		if(n%2==1) return nums1[n/2];
	    		else return 1.0*(nums1[n/2]+nums1[n/2-1])/2.0;	    		
	    	}
	    	int t=(m+n-1)/2;
	    	while((l1<r1-1)||(l2<r2-1)){
	    		int mid1 = l1+(r1-l1)/2;
	    		int mid2 = l2+(r2-l2)/2;
	    		int n1 = find(nums2, nums1[mid1]);
	    		int n2 = find(nums1, nums2[mid2]);
	    		Dumper.print(mid1+" "+mid2+" "+n1+" "+n2);
	    		if(n1+mid1==t){
	    			if((m+n)%2==1) return nums1[mid1];
	    			else return 1.0*(nums1[mid1]+nums2[n1])/2.0;
	    		}else if(n1+mid1>t){
	    			if(l2<r2-1) l2=mid2;//r2=Math.min(n1, r2);
	    			if(l1<r1-1) r1=mid1;//l1=Math.max(l1, n2);
	    		}else{
	    			if(l2<r2-1) r2=mid2;//l2=Math.max(n2, l2);
	    			if(l1<r1-1) l1=mid1;//r1=Math.min(r1, n2);
	    		}
	    	}
	    	int a=find(nums2,nums1[l1]);
	    	int b=find(nums2,nums1[r1]);
	    	int c=find(nums1,nums2[l2]);
	    	int d=find(nums1,nums2[r2]);
	    	if(a+l1==t){
    			if((m+n)%2==1) return nums1[l1];
    			else{
    				int tmp = nums1[r1];
    				if(nums2[l2]>=tmp&&nums2[l2]<nums1[r1]) tmp = nums2[l2];
    				return 1.0*(nums1[l1]+tmp)/2.0;
    			}
	    	}
	    	if(b+r1==t){
    			if((m+n)%2==1) return nums1[r1];
    			else{
    				int tmp = nums2[l2];
    				return 1.0*(nums1[r1]+tmp)/2.0;
    			}
	    	}
	    	if(c+l2==t){
    			if((m+n)%2==1) return nums2[l2];
    			else{
    				int tmp = nums2[r2];
    				if(nums1[l1]>=tmp&&nums1[l1]<nums2[r2]) tmp = nums1[l1];
    				return 1.0*(nums2[l2]+tmp)/2.0;
    			}
	    	}
	    	if(d+r2==t){
    			if((m+n)%2==1) return nums2[r2];
    			else{
    				int tmp = nums1[l1];
    				return 1.0*(nums2[r2]+tmp)/2.0;
    			}
	    	}	    	
	        return 1.0;
	    } 
	    int find(int arr[], int v){
	    	int l=0; int r=arr.length-1;
	    	while(l<r-1){
	    		int mid = l+(r-l)/2;
	    		if(arr[mid]<v){
	    			l=mid;
	    		}else{
	    			r=mid;
	    		}
	    	}
	    	if(arr[r]<=v) return r+1;
	    	if(arr[l]<=v) return l+1;
	    	return 0;
	    }
		
	    public List<Integer> fallingSquares(int[][] positions) {
	        int n = positions.length;
	        ArrayList<Integer> arr = new ArrayList<Integer>();
	        for(int i=0;i<n;i++){
	        	arr.add(positions[i][0]);
	        	arr.add(positions[i][0]+positions[i][1]-1);
	        }
	        Collections.sort(arr);	        
	        HashMap<Integer, Integer> mp = new HashMap<Integer,Integer>();
	        for(int i=0,j=0;i<arr.size();i++){
	        	if(!mp.containsKey(arr.get(i))){
	        		mp.put(arr.get(i),j++);
	        	}
	        }
	        int square[][] = new int[n][3];
	        for(int i=0;i<n;i++){
	        	square[i][0] = mp.get(positions[i][0]);
	        	square[i][1] = mp.get(positions[i][0]+positions[i][1]-1); 
	        	square[i][2] = positions[i][1];
	        }
	        List<Integer> result = new ArrayList<Integer>();
	        seg_tree t = new seg_tree(mp.size());
	        int tmp_v = 1;
    		while(tmp_v<mp.size()) {
    			tmp_v*=2;
    		}	        
	        for(int i=0;i<n;i++){
	        	int tmp = t.query1(square[i][0], square[i][1]+1, 0, 0, tmp_v);
	        	System.out.println(tmp);
	        	t.update(square[i][0], square[i][1]+1, 0, 0, tmp_v, square[i][2]+tmp);	        	
	        	tmp = t.query(0,tmp_v, 0, 0, tmp_v);
	        	result.add(tmp);
	        }
	        return result;
	    }
	    class seg_tree{
	    	int n;
	    	int arr_leaf[];
	    	int arr_node[];
	    	public seg_tree(int x) {
	    		n=1;	    		
	    		while(n<x) {
	    			n*=2;
	    		}
	    		arr_leaf = new int[2*n-1];
	    		arr_node = new int[2*n-1];
	    	}
	    	public int update(int a, int b, int k, int i, int j, int v) {
	    		if(a<=i&&j<=b) {
	    			arr_leaf[k]=v;	   
	    			return Math.max(arr_leaf[k], arr_node[k]);
	    		}else if(i<b&&a<j){
	    			int t1 = update(a,b,k*2+1,i,(i+j)/2,v);
	    			int t2 = update(a,b,k*2+2,(i+j)/2,j,v);
	    			arr_node[k] = Math.max(arr_node[k], Math.max(t1, t2));
	    			return arr_node[k];
	    		}else {
	    			return 0;
	    		}
	    	}
	    	public int query(int a, int b, int k, int i, int j) {
	    		if(b<=i||j<=a) {
	    			return 0;
	    		}else if(a<=i&&j<=b) {
	    			return Math.max(arr_leaf[k], arr_node[k]);
	    		}else {
	    			int t1 = query(a,b,k*2+1,i,(i+j)/2);
	    			int t2 = query(a,b,k*2+2,(i+j)/2,j);
	    			return Math.max(t1,t2);
	    		}	    		
	    	}
	    	public int query1(int a, int b, int k, int i, int j) {
	    		if(b<=i||j<=a) {
	    			return 0;
	    		}else if(a<=i&&j<=b) {
	    			return Math.max(arr_leaf[k], arr_node[k]);
	    		}else {
	    			//if(arr_leaf[k]!=0) return arr_leaf[k];
	    			int t1 = query1(a,b,k*2+1,i,(i+j)/2);
	    			int t2 = query1(a,b,k*2+2,(i+j)/2,j);
	    			return Math.max(arr_leaf[k], Math.max(t1,t2));
	    		}	    		
	    	}	    	
	    }
	    
		public ListNode reverseKGroup(ListNode head, int k) {
            if(k<=1||head==null||head.next==null) return head; 
			ListNode dummy = new ListNode(0);
			dummy.next = head;
			ListNode pre1 = dummy;
			ListNode pre2 = pre1;
			ListNode p = head;
			ListNode q = p;
			while(true){
				int c=1;
				boolean flag = true;				
				while(q!=null&&q.next!=null&&c<k){
					q = q.next;
					c++;                
					flag = false;                 
				}			
				if(c<k||flag) break;
				int d = k;	
				c=1;
				q=p;
				ListNode end = q;
				while(c<d&&d>1){
					c=1;
					while(c<d){
						q=q.next;
						pre2=pre2.next;
						c++;
					}
					//System.out.println(p.val+" "+q.val);	
					
					if(d==2){
						p.next=q.next;
						q.next=p;
						pre1.next=q;
					}else{
						ListNode tmp = p.next;
						p.next=q.next;
						q.next=tmp;
						pre1.next=q;
						pre2.next=p;
					}
					Dumper.print_listnode(dummy);					
					if(d>3){
						p=q.next;	
						pre1=q;											
					}else{
						pre1=end;
						p=end.next;						
					}
					q=p;
					pre2=pre1;
					c=0;
					d-=2;					
				}				
			}
			return dummy.next;
		}
		
		ListNode create(int arr[]){
			int p=1;
			ListNode head = new ListNode(arr[0]);
			ListNode pre = head;
			while(p<arr.length){
				ListNode tmp = new ListNode(arr[p++]);
				pre.next=tmp;
				pre=pre.next;
			}
			return head;
		}
        
        public int[] maxNumber1(int[] nums1, int[] nums2, int k) {
            int m = nums1.length;
            int n = nums2.length;
            int arr1[] = new int[m];
            int arr2[] = new int[n];
            for(int i=0;i<m;i++) arr1[i] = nums1[i];
            for(int i=0;i<n;i++) arr2[i] = nums2[i];  
            String dp[][][] = new String[m+1][n+1][k+1];
            for(int i=0;i<=m;i++){
            	for(int j=0;j<=n;j++){
            		for(int p=0;p<=k;p++){
            			dp[i][j][p] = "";
            		}
            	}
            }
            for(int i=0;i<=m;i++){
            	for(int j=0;j<=n;j++){
            		for(int p=1;p<=Math.min(i+j, k);p++){
            			if(i>=1&&dp[i-1][j][p].compareTo(dp[i][j][p])>0) {
            				dp[i][j][p] = dp[i-1][j][p];
            			}
            			if(j>=1&&dp[i][j-1][p].compareTo(dp[i][j][p])>0) {
            				dp[i][j][p] = dp[i][j-1][p];
            			}
            			String t1 = "";
            			String t2 = "";
            			if(i>=1) t1 = dp[i-1][j][p-1]+arr1[i-1];
            			if(j>=1) t2 = dp[i][j-1][p-1]+arr2[j-1];
            			if(t1.compareTo((dp[i][j][p]))>0) {
            				dp[i][j][p] = t1;
            			}
            			if(t2.compareTo((dp[i][j][p]))>0) {
            				dp[i][j][p] = t2;
            			}
            		}
            	}
            }
            int result[] = new int[k];
            char tmp[] = dp[m][n][k].toCharArray();
            for(int i=0;i<k;i++){
            	result[i] = tmp[i]-'0';
            }
            return result;
        }

		class TreeNode {
			int val;
			TreeNode left;
			TreeNode right;

			public TreeNode(int a) {
				val = a;
			}
		}

		public class Interval {
			int start;
			int end;

			Interval() {
				start = 0;
				end = 0;
			}

			Interval(int s, int e) {
				start = s;
				end = e;
			}
		}
	}

	// default class
	static class InputReader {
		public BufferedReader re;
		public StringTokenizer st;

		public InputReader(InputStream input) {
			re = new BufferedReader(new InputStreamReader(input));
			st = null;
		}

		public String next() {
			while (st == null || !st.hasMoreTokens()) {
				try {
					st = new StringTokenizer(re.readLine());
				} catch (IOException e) {
					throw new RuntimeException(e);
				}

			}
			return st.nextToken();
		}

		public String nextLine() {
			String tmp = null;
			try {
				tmp = re.readLine();
			} catch (IOException e) {
				throw new RuntimeException(e);
			}
			return tmp;
		}

		public boolean hasNext() {
			String tmp;
			if (st != null && st.hasMoreTokens())
				return true;
			else {
				try {
					tmp = re.readLine();
				} catch (IOException e) {
					throw new RuntimeException(e);
				}
				if (tmp != null) {
					st = new StringTokenizer(tmp);
					return true;
				} else {
					return false;
				}
			}

		}

		public int nextInt() {
			return Integer.parseInt(next());
		}

		public double nextDouble() {
			return Double.parseDouble(next());
		}

		public long nextLong() {
			return Long.parseLong(next());
		}
	}

	static class Dumper {
		static void print_int_arr(int[] arr) {
			for (int i = 0; i < arr.length; i++) {
				System.out.print(arr[i] + " ");
			}
			System.out.println();
			System.out.println("---------------------");
		}
		
		static void print_char_arr(char[] arr) {
			for (int i = 0; i < arr.length; i++) {
				System.out.print(arr[i] + " ");
			}
			System.out.println();
			System.out.println("---------------------");
		}		

		static void print_double_arr(double[] arr) {
			for (int i = 0; i < arr.length; i++) {
				System.out.print(arr[i] + " ");
			}
			System.out.println();
			System.out.println("---------------------");
		}

		static void print_2d_arr(int[][] arr, int x, int y) {
			for (int i = 0; i < x; i++) {
				for (int j = 0; j < y; j++) {
					System.out.print(arr[i][j] + " ");
				}
				System.out.println();
			}
			System.out.println();
			System.out.println("---------------------");
		}

		static void print_2d_arr(boolean[][] arr, int x, int y) {
			for (int i = 0; i < x; i++) {
				for (int j = 0; j < y; j++) {
					System.out.print(arr[i][j] + " ");
				}
				System.out.println();
			}
			System.out.println();
			System.out.println("---------------------");
		}

		static void print(Object o) {
			System.out.println(o.toString());
		}

		static void getc() {
			System.out.println("here");
		}
		
		static void print_listnode(ListNode x){
			while(x!=null){
				System.out.print(x.val+" ");
				x=x.next;
			}
			System.out.println();
		}
	}
}