USACO 2025 February Contest, Bronze

Problem 3. Printing Sequences


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Bessie is learning to code using a simple programming language. She first defines a valid program, then executes it to produce some output sequence.

Defining:

  • A program is a nonempty sequence of statements.
  • A statement is either of the form "PRINT $c$" where $c$ is an integer, or "REP $o$", followed by a program, followed by "END," where $o$ is an integer that is at least 1.
Executing:
  • Executing a program executes its statements in sequence.
  • Executing the statement "PRINT $c$" appends $c$ to the output sequence.
  • Executing a statement starting with "REP $o$" executes the inner program a total of $o$ times in sequence.

An example of a program that Bessie knows how to write is as follows. 

REP 3
    PRINT 1
    REP 2
        PRINT 2
    END
END

The program outputs the sequence $[1,2,2,1,2,2,1,2,2]$.

Bessie wants to output a sequence of $N$ ($1 \le N \le 100$) positive integers. Elsie challenges her to use no more than $K$ ($1 \le K \le 3$) "PRINT" statements. Note that Bessie can use as many "REP" statements as she wants. Also note that each positive integer in the sequence is no greater than $K$.

For each of $T$ ($1 \le T \le 100$) independent test cases, determine whether Bessie can write a program that outputs some given sequence using at most $K$ "PRINT" statements.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $T$.

The first line of each test case contains two space-separated integers, $N$ and $K$.

The second line of each test case contains a sequence of $N$ space-separated positive integers, each at most $K$, which is the sequence that Bessie wants to produce.

OUTPUT FORMAT (print output to the terminal / stdout):

For each test case, output "YES" or "NO" (case sensitive) on a separate line.

SAMPLE INPUT:

2
1 1
1
4 1
1 1 1 1

SAMPLE OUTPUT:

YES
YES
For the second test case, the following code outputs the sequence $[1,1,1,1]$ with $1$ "PRINT" statement. 

REP 4
    PRINT 1
END

SAMPLE INPUT:

11
4 2
1 2 2 2
4 2
1 1 2 1
4 2
1 1 2 2
6 2
1 1 2 2 1 1
10 2
1 1 1 2 2 1 1 1 2 2
8 3
3 3 1 2 2 1 2 2
9 3
1 1 2 2 2 3 3 3 3
16 3
2 2 3 2 2 3 1 1 2 2 3 2 2 3 1 1
24 3
1 1 2 2 3 3 3 2 2 3 3 3 1 1 2 2 3 3 3 2 2 3 3 3
9 3
1 2 2 1 3 3 1 2 2
6 3
1 2 1 2 2 3

SAMPLE OUTPUT:

YES
NO
YES
NO
YES
YES
YES
YES
YES
NO
NO
For the first test case, the following code outputs the sequence $[1,2,2,2]$ with $2$ "PRINT" statements. 

PRINT 1
REP 3
    PRINT 2
END

For the second test case, the answer is "NO" because it is impossible to output the sequence $[1,1,2,1]$ using at most $2$ "PRINT" statements.

For the sixth test case, the following code outputs the sequence $[3,3,1,2,2,1,2,2]$ with $3$ "PRINT" statements. 

REP 2
    PRINT 3
END
REP 2
    PRINT 1
    REP 2
        PRINT 2
    END
END

SCORING:

  • Input 3: $K=1$
  • Inputs 4-7: $K \le 2$
  • Inputs 8-13: No additional constraints.

Problem credits: Alex Liang

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