apaz/codeforces-genesys huggingface dataset:
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
Define the range of a non-empty array to be the maximum value minus the minimum value. For example, the range of $$$[1,4,2]$$$ is $$$4-1=3$$$.

You are given an array $$$a_1, a_2, \ldots, a_n$$$ of length $$$n \geq 3$$$. It is guaranteed $$$a$$$ is sorted.

You have to color each element of $$$a$$$ red or blue so that:

- the range of the red elements does not equal the range of the blue elements, and
- there is at least one element of each color.

If there does not exist any such coloring, you should report it. If there are multiple valid colorings, you can print any of them.

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 100$$$) — the number of test cases.

The first line of each test case contains an integer $$$n$$$ ($$$3 \leq n \leq 50$$$) — the length of the array.

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \leq a_i \leq 10^9$$$). It is guaranteed $$$a_1 \leq a_2 \leq \ldots \leq a_{n - 1} \leq a_{n}$$$.

## Output Format
For each test case, if it is impossible to color $$$a$$$ to satisfy all the constraints, output $$$\texttt{NO}$$$.

Otherwise, first output $$$\texttt{YES}$$$.

Then, output a string $$$s$$$ of length $$$n$$$. For $$$1 \leq i \leq n$$$, if you color $$$a_i$$$ red, $$$s_i$$$ should be $$$\texttt{R}$$$. For $$$1 \leq i \leq n$$$, if you color $$$a_i$$$ blue, $$$s_i$$$ should be $$$\texttt{B}$$$.

## Examples
```input
7
4
1 1 2 2
5
1 2 3 4 5
3
3 3 3
4
1 2 2 2
3
1 2 2
3
1 1 2
3
1 9 84
```
```output
YES
RBRR
YES
BBRBB
NO
YES
RBBR
YES
RRB
YES
BRR
YES
BRB
```

## Note
In the first test case, given the array $$$[1, 1, 2, 2]$$$, we can color the second element blue and the remaining elements red; then the range of the red elements $$$[1, 2, 2]$$$ is $$$2-1=1$$$, and the range of the blue elements $$$[1]$$$ is $$$1-1=0$$$.

In the second test case, we can color the first, second, fourth and fifth elements $$$[1, 2, 4, 5]$$$ blue and the remaining elements $$$[3]$$$ red.

The range of the red elements is $$$3 - 3 = 0$$$ and the range of the blue elements is $$$5 - 1 = 4$$$, which are different.

In the third test case, it can be shown there is no way to color $$$a = [3, 3, 3]$$$ to satisfy the constraints.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
A digit is large if it is between $$$5$$$ and $$$9$$$, inclusive. A positive integer is large if all of its digits are large.

You are given an integer $$$x$$$. Can it be the sum of two large positive integers with the same number of digits?

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases.

The only line of each test case contains a single integer $$$x$$$ ($$$10 \leq x \leq 10^{18}$$$).

## Output Format
For each test case, output $$$\texttt{YES}$$$ if $$$x$$$ satisfies the condition, and $$$\texttt{NO}$$$ otherwise.

You can output $$$\texttt{YES}$$$ and $$$\texttt{NO}$$$ in any case (for example, strings $$$\texttt{yES}$$$, $$$\texttt{yes}$$$, and $$$\texttt{Yes}$$$ will be recognized as a positive response).

## Examples
```input
11
1337
200
1393938
1434
98765432123456789
11111111111111111
420
1984
10
69
119
```
```output
YES
NO
YES
YES
NO
YES
NO
YES
YES
NO
NO
```

## Note
In the first test case, we can have $$$658 + 679 = 1337$$$.

In the second test case, it can be shown that no numbers of equal length and only consisting of large digits can add to $$$200$$$.

In the third test case, we can have $$$696\,969 + 696\,969 = 1\,393\,938$$$.

In the fourth test case, we can have $$$777 + 657 = 1434$$$.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
The two versions of the problem are different. You may want to read both versions. You can make hacks only if both versions are solved.

You are given an array $$$a$$$ of length $$$n$$$. Start with $$$c = 0$$$. Then, for each $$$i$$$ from $$$1$$$ to $$$n$$$ (in increasing order) do exactly one of the following:

- Option $$$1$$$: set $$$c$$$ to $$$c + a_i$$$.
- Option $$$2$$$: set $$$c$$$ to $$$|c + a_i|$$$, where $$$|x|$$$ is the absolute value of $$$x$$$.

Let the maximum final value of $$$c$$$ after the procedure described above be equal to $$$k$$$. Find the number of unique procedures that result in $$$c = k$$$. Two procedures are different if at any index $$$i$$$, one procedure chose option $$$1$$$ and another chose option $$$2$$$, even if the value of $$$c$$$ is equal for both procedures after that turn.

Since the answer may be large, output it modulo $$$998\,244\,353$$$.

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 2 \cdot 10^5$$$).

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$-10^9 \leq a_i \leq 10^9$$$).

The sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

## Output Format
For each test case, output a single integer — the number of unique procedures that result in $$$c = k$$$, modulo $$$998\,244\,353$$$.

## Examples
```input
5
4
2 -5 3 -3
8
1 4 3 4 1 4 3 4
3
-1 -2 -3
4
-1000000000 1000000000 1000000000 1000000000
4
1 9 8 4
```
```output
12
256
1
8
16
```

## Note
In the first test case, it can be shown that our maximal final value of $$$c$$$ is $$$3$$$. There are $$$12$$$ ways to achieve this because in order to get $$$3$$$, we have to take absolute value at indices $$$2$$$ or $$$4$$$, or both, resulting in $$$3$$$ ways. For the other two indices, it doesn't change the value whether we take absolute value or not, so we have $$$2 \cdot 2 = 4$$$ ways for them. In total, we have $$$3 \cdot 4 = 12$$$ ways.

In the second test case, taking the absolute value will never change anything, so we can either take absolute value or not, for every index. This gives us $$$2^8 = 256$$$ possible ways.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
You are given a string $$$s$$$ consisting of lowercase Latin characters. Count the number of nonempty strings $$$t \neq$$$ "$$$\texttt{a}$$$" such that it is possible to partition$$$^{\dagger}$$$ $$$s$$$ into some substrings satisfying the following conditions:

- each substring either equals $$$t$$$ or "$$$\texttt{a}$$$", and
- at least one substring equals $$$t$$$.

$$$^{\dagger}$$$ A partition of a string $$$s$$$ is an ordered sequence of some $$$k$$$ strings $$$t_1, t_2, \ldots, t_k$$$ (called substrings) such that $$$t_1 + t_2 + \ldots + t_k = s$$$, where $$$+$$$ represents the concatenation operation.

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases.

The only line of each test case contains a string $$$s$$$ consisting of lowercase Latin characters ($$$2 \leq |s| \leq 2 \cdot 10^5$$$).

The sum of $$$|s|$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

## Output Format
For each test case, output a single integer — the number of nonempty strings $$$t \neq$$$ "$$$\texttt{a}$$$" that satisfy all constraints.

## Examples
```input
8
aaaaa
baba
cabacb
aaabaaa
bitset
ab
abbaaaabbb
yearnineteeneightyfour
```
```output
4
4
1
16
1
2
3
1
```

## Note
In the first test case, $$$t$$$ can be "$$$\texttt{aa}$$$", "$$$\texttt{aaa}$$$", "$$$\texttt{aaaa}$$$", or the full string.

In the second test case, $$$t$$$ can be "$$$\texttt{b}$$$", "$$$\texttt{bab}$$$", "$$$\texttt{ba}$$$", or the full string.

In the third test case, the only such $$$t$$$ is the full string.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
The secret behind Oscar's first magic trick has been revealed! Because he still wants to impress Lura, he comes up with a new idea: he still wants to sort a permutation $$$p_1, p_2, \ldots, p_n$$$ of $$$[1, 2, \ldots, n]$$$.

This time, he chooses an integer $$$k$$$. He wants to sort the permutation in non-decreasing order using the following operation several times:

1. Pick a continuous subarray of length $$$k$$$ and remove it from $$$p$$$.
2. Insert the continuous subarray back into $$$p$$$ at any position (perhaps, in the very front or the very back).

To be as impressive as possible, Oscar would like to choose the maximal value of $$$k$$$ such that he can sort his permutation. Please help him find the maximal $$$k$$$ as well as a sequence of operations that will sort the permutation. You don't need to minimize the number of operations, but you are allowed to use at most $$$5n^2$$$ operations.

We have a proof that, for the maximal $$$k$$$ such that you can sort the permutation in any number of operations, you can also sort it in at most $$$5n^2$$$ operations.

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases.

The first line of each test case contains a single integer $$$n$$$ ($$$5 \leq n \leq 10^3$$$) — the length of the permutation.

The second line of each test case contains a permutation $$$p_1, p_2, \ldots, p_n$$$ of $$$[1, 2, \ldots, n]$$$.

The sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^3$$$.

## Output Format
For each test case, first output the chosen value of $$$k$$$ on a new line ($$$1 \leq k \leq n$$$).

Then, output a single integer $$$m$$$ — the number of operations used ($$$0 \leq m \leq 5n^2$$$).

Then, on each of the next $$$m$$$ lines, output the operations denoted by two integers $$$i$$$ and $$$j$$$ ($$$1 \leq i, j \leq n - k + 1$$$), representing an operation where you remove the subarray starting from index $$$i$$$ and replace it back into $$$p$$$ at index $$$j$$$.

## Examples
```input
3
5
5 1 2 3 4
5
2 3 5 4 1
6
1 2 3 4 5 6
```
```output
4
1
2 1
3
2
1 3
2 1
6
0
```

## Note
In the first test case, it is enough to move the last four numbers to the front.

In the second test case, it can be shown that we cannot have $$$k = 4$$$ or $$$k = 5$$$. With $$$k = 3$$$, we can move the first three numbers to the end, and then the middle three to the front to sort the permutation.

In the third test case, the permutation is already sorted. We can have $$$k = 6$$$ and use no operations.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
There are $$$n$$$ towers at $$$n$$$ distinct points $$$(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$$$, such that no three are collinear and no four are concyclic. Initially, you own towers $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$, and you want to capture all of them. To do this, you can do the following operation any number of times:

- Pick two towers $$$P$$$ and $$$Q$$$ you own and one tower $$$R$$$ you don't own, such that the circle through $$$P$$$, $$$Q$$$, and $$$R$$$ contains all $$$n$$$ towers inside of it.
- Afterwards, capture all towers in or on triangle $$$\triangle PQR$$$, including $$$R$$$ itself.

Count the number of attack plans of minimal length. Note that it might not be possible to capture all towers, in which case the answer is $$$0$$$.

Since the answer may be large, output it modulo $$$998\,244\,353$$$.

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 250$$$) — the number of test cases.

The first line of each test case contains a single integer $$$n$$$ ($$$4 \leq n \leq 100$$$) — the number of towers.

The $$$i$$$-th of the next $$$n$$$ lines contains two integers $$$x_i$$$ and $$$y_i$$$ ($$$-10^4 \leq x_i, y_i \leq 10^4$$$) — the location of the $$$i$$$-th tower. Initially, you own towers $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$.

All towers are at distinct locations, no three towers are collinear, and no four towers are concyclic.

The sum of $$$n$$$ over all test cases does not exceed $$$1000$$$.

## Output Format
For each test case, output a single integer — the number of attack plans of minimal length after which you capture all towers, modulo $$$998\,244\,353$$$.

## Examples
```input
3
5
1 1
2 5
3 3
4 2
5 4
6
1 1
3 3
1 2
2 1
3 10000
19 84
7
2 7
-4 -3
-3 6
3 1
-5 2
1 -4
-1 7
```
```output
1
0
10
```

## Note
In the first test case, there is only one possible attack plan of shortest length, shown below.

- Use the operation with $$$P =$$$ tower $$$1$$$, $$$Q =$$$ tower $$$2$$$, and $$$R =$$$ tower $$$5$$$. The circle through these three towers contains all towers inside of it, and as a result towers $$$3$$$ and $$$5$$$ are captured.
- Use the operation with $$$P =$$$ tower $$$5$$$, $$$Q =$$$ tower $$$1$$$, and $$$R =$$$ tower $$$4$$$. The circle through these three towers contains all towers inside of it, and as a result tower $$$4$$$ is captured.

In the second case, for example, you can never capture the tower at $$$(3, 10\,000)$$$.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.5 seconds
Memory limit: 512.0 MB

# Problem
You are given a sequence $$$[a_1,\ldots,a_n]$$$ where each element $$$a_i$$$ is either $$$0$$$ or $$$1$$$. You can apply several (possibly zero) operations to the sequence. In each operation, you select two integers $$$1\le l\le r\le |a|$$$ (where $$$|a|$$$ is the current length of $$$a$$$) and replace $$$[a_l,\ldots,a_r]$$$ with a single element $$$x$$$, where $$$x$$$ is the majority of $$$[a_l,\ldots,a_r]$$$.

Here, the majority of a sequence consisting of $$$0$$$ and $$$1$$$ is defined as follows: suppose there are $$$c_0$$$ zeros and $$$c_1$$$ ones in the sequence, respectively.

- If $$$c_0\ge c_1$$$, the majority is $$$0$$$.
- If $$$c_0<c_1$$$, the majority is $$$1$$$.

For example, suppose $$$a=[1,0,0,0,1,1]$$$. If we select $$$l=1,r=2$$$, the resulting sequence will be $$$[0,0,0,1,1]$$$. If we select $$$l=4,r=6$$$, the resulting sequence will be $$$[1,0,0,1]$$$.

Determine if you can make $$$a=[1]$$$ with a finite number of operations.

## Input Format
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 4\cdot 10^4$$$). Description of the test cases follows.

The first line of each testcase contains one integer $$$n$$$ ($$$1\le n\le 2\cdot 10^5$$$).

The second line of each testcase contains a string consisting of $$$0$$$ and $$$1$$$, describing the sequence $$$a$$$.

It's guaranteed that the sum of $$$n$$$ over all testcases does not exceed $$$2\cdot 10^5$$$.

## Output Format
For each testcase, if it's possible to make $$$a=[1]$$$, print YES. Otherwise, print NO. You can output the answer in any case (upper or lower). For example, the strings yEs, yes, Yes, and YES will be recognized as positive responses.

## Examples
```input
5
1
0
1
1
2
01
9
100000001
9
000011000
```
```output
No
Yes
No
Yes
No
```

## Note
In the fourth testcase of the example, initially $$$a=[1,0,0,0,0,0,0,0,1]$$$. A valid sequence of operations is:

1. Select $$$l=2,r=8$$$ and apply the operation. $$$a$$$ becomes $$$[1,0,1]$$$.
2. Select $$$l=1,r=3$$$ and apply the operation. $$$a$$$ becomes $$$[1]$$$.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 512.0 MB

# Problem
You are given a positive integer $$$n$$$. Find the longest sequence of positive integers $$$a=[a_1,a_2,\ldots,a_k]$$$ that satisfies the following conditions, and print the sequence:

- $$$a_i\le n$$$ for all $$$1\le i\le k$$$.
- $$$a$$$ is strictly increasing. That is, $$$a_i>a_{i-1}$$$ for all $$$2\le i\le k$$$.
- $$$a_i\,|\,a_{i-1}=n$$$ for all $$$2\le i\le k$$$, where $$$|$$$ denotes the bitwise OR operation.

## Input Format
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 1000$$$). Description of the test cases follows.

The only line of each test case contains one integer $$$n$$$ ($$$1\le n\le 10^{18}$$$).

It's guaranteed that the sum of lengths of the longest valid sequences does not exceed $$$5\cdot 10^5$$$.

## Output Format
For each testcase, print two lines. In the first line, print the length of your constructed sequence, $$$k$$$. In the second line, print $$$k$$$ positive integers, denoting the sequence. If there are multiple longest sequences, you can print any of them.

## Examples
```input
4
1
3
14
23
```
```output
1
1
3
1 2 3
4
4 10 12 14
5
7 18 21 22 23
```

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 5.0 seconds
Memory limit: 512.0 MB

# Problem
For an array $$$u_1, u_2, \ldots, u_n$$$, define

- a prefix maximum as an index $$$i$$$ such that $$$u_i>u_j$$$ for all $$$j<i$$$;
- a suffix maximum as an index $$$i$$$ such that $$$u_i>u_j$$$ for all $$$j>i$$$;
- an ascent as an index $$$i$$$ ($$$i>1$$$) such that $$$u_i>u_{i-1}$$$.

You are given three cost arrays: $$$[a_1, a_2, \ldots, a_n]$$$, $$$[b_1, b_2, \ldots, b_n]$$$, and $$$[c_0, c_1, \ldots, c_{n-1}]$$$.

Define the cost of an array that has $$$x$$$ prefix maximums, $$$y$$$ suffix maximums, and $$$z$$$ ascents as $$$a_x\cdot b_y\cdot c_z$$$.

Let the sum of costs of all permutations of $$$1,2,\ldots,n$$$ be $$$f(n)$$$. Find $$$f(1)$$$, $$$f(2)$$$, ..., $$$f(n)$$$ modulo $$$998\,244\,353$$$.

## Input Format
The first line contains an integer $$$n$$$ ($$$1\le n\le 700$$$).

The second line contains $$$n$$$ integers $$$a_1,\ldots,a_n$$$ ($$$0\le a_i<998\,244\,353$$$).

The third line contains $$$n$$$ integers $$$b_1,\ldots,b_n$$$ ($$$0\le b_i<998\,244\,353$$$).

The fourth line contains $$$n$$$ integers $$$c_0,\ldots,c_{n-1}$$$ ($$$0\le c_i<998\,244\,353$$$).

## Output Format
Print $$$n$$$ integers: the $$$i$$$-th one is $$$f(i)$$$ modulo $$$998\,244\,353$$$.

## Examples
```input
3
1 1 1
1 1 1
1 1 1
```
```output
1 2 6
```
-----
```input
3
1 2 3
2 3 1
3 1 2
```
```output
6 13 34
```
-----
```input
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
```
```output
600000000 303511294 612289529 324650937 947905622
```

## Note
In the second example:

- Consider permutation $$$[1,2,3]$$$. Indices $$$1,2,3$$$ are prefix maximums. Index $$$3$$$ is the only suffix maximum. Indices $$$2,3$$$ are ascents. In conclusion, it has $$$3$$$ prefix maximums, $$$1$$$ suffix maximums, and $$$2$$$ ascents. Therefore, its cost is $$$a_3b_1c_2=12$$$.
- Permutation $$$[1,3,2]$$$ has $$$2$$$ prefix maximums, $$$2$$$ suffix maximums, and $$$1$$$ ascent. Its cost is $$$6$$$.
- Permutation $$$[2,1,3]$$$ has $$$2$$$ prefix maximums, $$$1$$$ suffix maximum, and $$$1$$$ ascent. Its cost is $$$4$$$.
- Permutation $$$[2,3,1]$$$ has $$$2$$$ prefix maximums, $$$2$$$ suffix maximums, and $$$1$$$ ascent. Its cost is $$$6$$$.
- Permutation $$$[3,1,2]$$$ has $$$1$$$ prefix maximum, $$$2$$$ suffix maximums, and $$$1$$$ ascent. Its cost is $$$3$$$.
- Permutation $$$[3,2,1]$$$ has $$$1$$$ prefix maximum, $$$3$$$ suffix maximums, and $$$0$$$ ascents. Its cost is $$$3$$$.

The sum of all permutations' costs is $$$34$$$, so $$$f(3)=34$$$.

Now solve the problem and return the code.