Carousel
Contest: Codeforces Round 629 (Div. 3)
Difficulty rating: 1800
Tags: constructive algorithms, dp, graphs, greedy, math

The round carousel consists of $$$n$$$ figures of animals. Figures are numbered from $$$1$$$ to $$$n$$$ in order of the carousel moving. Thus, after the $$$n$$$-th figure the figure with the number $$$1$$$ follows. Each figure has its own type — the type of the animal corresponding to this figure (the horse, the tiger and so on). The type of animal of the $$$i$$$-th figure equals $$$t_i$$$.

The example of the carousel for $$$n=9$$$ and $$$t=[5, 5, 1, 15, 1, 5, 5, 1, 1]$$$.

You want to color each figure in one of the colors. You think that it's boring if the carousel contains two different figures (with the distinct types of animals) going one right after another and colored in the same color.

Your task is to color the figures in such a way that the number of distinct colors used is the minimum possible and there are no figures of the different types going one right after another and colored in the same color. If you use exactly $$$k$$$ distinct colors, then the colors of figures should be denoted with integers from $$$1$$$ to $$$k$$$.

Input format:
The input contains one or more test cases.

The first line contains one integer $$$q$$$ ($$$1 \le q \le 10^4$$$) — the number of test cases in the test. Then $$$q$$$ test cases follow. One test case is given on two lines.

The first line of the test case contains one integer $$$n$$$ ($$$3 \le n \le 2 \cdot 10^5$$$) — the number of figures in the carousel. Figures are numbered from $$$1$$$ to $$$n$$$ in order of carousel moving. Assume that after the $$$n$$$-th figure the figure $$$1$$$ goes.

The second line of the test case contains $$$n$$$ integers $$$t_1, t_2, \dots, t_n$$$ ($$$1 \le t_i \le 2 \cdot 10^5$$$), where $$$t_i$$$ is the type of the animal of the $$$i$$$-th figure.

The sum of $$$n$$$ over all test cases does not exceed $$$2\cdot10^5$$$.

Output format:
Print $$$q$$$ answers, for each test case print two lines.

In the first line print one integer $$$k$$$ — the minimum possible number of distinct colors of figures.

In the second line print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$1 \le c_i \le k$$$), where $$$c_i$$$ is the color of the $$$i$$$-th figure. If there are several answers, you can print any.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Example input:
4
5
1 2 1 2 2
6
1 2 2 1 2 2
5
1 2 1 2 3
3
10 10 10

Example output:
2
1 2 1 2 2
2
2 1 2 1 2 1
3
2 3 2 3 1
1
1 1 1

Editorial:
The answer to this problem is at most $$$3$$$. Let's prove it by construction.
Firstly, if all $$$t_i$$$ are equal then the answer is $$$1$$$. Otherwise, there are at least two different values in the array $$$t$$$ so the answer is at least $$$2$$$. If $$$n$$$ is even then the answer is always $$$2$$$ because you can color figures in the following way: $$$[1, 2, 1, 2, \dots, 1, 2]$$$. If $$$n$$$ is odd then consider two cases. The first case is when some pair of adjacent figures have the same type. Then the answer is $$$2$$$ because you can merge these two values into one and get the case of even $$$n$$$. Otherwise, all pairs of adjacent figures have different types and if you consider this cyclic array as a graph (cycle of length $$$n$$$) then you can notice that it isn't bipartite so you need at least $$$3$$$ colors to achieve the answer (color all vertices in such a way that any two adjacent vertices have different colors). And the answer looks like $$$[1, 2, 1, 2, \dots, 1, 2, 3]$$$.