Problem Description

Little T will give you an integer sequence of length $n$. You have a number $x$ in your hand, initially $0$. You can perform the following operations so that, in the end, the difference between $x$ and $c$ is less than $10^{-4}$.

You may perform at most $k$ operations on $x$:

• add i: add $a_i$ to the counter $x$, and then $a_i$ cannot be used in any operation anymore.

• sub i: subtract $a_i$ from the counter $x$, and then $a_i$ cannot be used in any operation anymore.

• mul i: multiply the counter $x$ by $a_i$, and then $a_i$ cannot be used in any operation anymore.

• sqrt i: set $a_i$ to $\sqrt {a_i}$. Each $a_i$ can be square-rooted at most once.

• pow f: change the counter $x$ into $x^f$, where $f$ can be a floating-point number.

All $a_i$ must be used exactly once in an addition, subtraction, or multiplication operation on $x$.

During the computation, the values of $a_i$ and $x$ must not exceed $10^{10}$. The problem guarantees that a solution exists. If there are multiple solutions, output any one of them.

This problem requires high precision. Please improve the precision of your algorithm.

Constraints

This problem uses bundled testdata.

• Subtask 0 (10 pts): $n \leq 5$, $k = n^2$. It is guaranteed that a solution can be obtained using only addition and subtraction.

• Subtask 1 (20 pts): $n \leq 5$, $k = n^2$. It is guaranteed that a solution can be obtained using addition, subtraction, multiplication, and square root.

• Subtask 2 (15 pts): $n \leq 10$, $a_i \leq 2$, $k = n + 1$.

• Subtask 3 (55 pts): $k = n + 1$.

For all testdata: $0 \leq n \leq 10^{5}$, $\sum_{i=1}^{n}{a_i} \le 10^{10}$, $0 \leq c \leq 10^{10}$.

Input Format

The first line contains three integers $n$, $k$, $c$.

The second line contains $n$ integers, representing the sequence $a$.

Output Format

The first line contains an integer $g$, the total number of operations.

The next $g$ lines contain your sequence of operations.

5 25 3
3 3 3 3 3

5
add 1
add 2
sub 3
sub 4
add 5

3 9 3
1 3 3

5
sqrt 2
sqrt 3
add 1
mul 2
mul 3

3 9 77
4 5 4

4
add 1
add 2
pow 2
sub 3

Hint

[Sample Explanation #1]

• Add $a_1$ to $x$. Now $x$ is $3$.

• Add $a_2$ to $x$. Now $x$ is $6$.

• Subtract $a_3$ from $x$. Now $x$ is $3$.

• Subtract $a_4$ from $x$. Now $x$ is $0$.

• Add $a_5$ to $x$. Now $x$ is $3$.

[Sample Explanation #2]

• Take the square root of $a_2$. Now $a = [1,\sqrt3,3]$.

• Take the square root of $a_3$. Now $a = [1,\sqrt3,\sqrt3]$.

• Add $a_1$ to $x$. Now $x$ is $1$.

• Multiply $x$ by $a_2$. Now $x$ is $\sqrt3$.

• Multiply $x$ by $a_3$. Now $x$ is $3$.

[Sample Explanation #3]

• Add $a_1$ to $x$. Now $x$ is $4$.

• Add $a_2$ to $x$. Now $x$ is $9$.

• Change $x$ into $x^2$. Now $x$ is $81$.

• Subtract $a_3$ from $x$. Now $x$ is $77$.

Translated by ChatGPT 5