A lock in the form of a mini-game#4651

A lock in the form of a mini-game

1 second

256 megabytes

Hard

In many computer games, lock picking is implemented as mini-games. Polycarp has designed a mini-game for his game with the following rules:

On the screen, three non-negative integers $a$ , $b$ , and $c$ are shown, as well as three buttons labeled "a", "b", and "c".

Button "a": Increases $a$ by $1$ and decreases $b$ and $c$ by $1$ each. This button may be pressed only if both $b$ and $c$ are greater than $0$ .
Button "b": Increases $b$ by $1$ and decreases $a$ and $c$ by $1$ each. This button may be pressed only if both $a$ and $c$ are greater than $0$ .
Button "c": Increases $c$ by $1$ and decreases $a$ and $b$ by $1$ each. This button may be pressed only if both $a$ and $b$ are greater than $0$ .

The lock is considered open if, after a sequence of button presses (possibly empty), the sum $a + b + c = 1$ a + b + c = 1.

Your task is to determine the minimum number of button presses required to open the lock, or determine if it is impossible. If the lock can be opened, output any optimal sequence of button presses that achieves this.

Input

The first line contains three non-negative integers $a$ , $b$ , $c$ ( $0 \leq a, b, c \leq 1 0^{9}$ 0 \le a, b, c \le 10^9, $1 \leq a + b + c \leq 1 0^{9}$ 1 \le a + b + c \le 10^9) — the initial numbers shown on the screen.

Output

Print one integer — the minimum number of presses required to open the lock. If it is impossible to open the lock, print $- 1$ .

If the lock can be opened, print on the second line any optimal sequence of presses — a string consisting only of the characters "a", "b", "c", "[", "]", and digits. The string must satisfy the following grammar (with start symbol answer):

character ::= a ∣ b ∣ c; sequence ::= character ∣ character sequence; block ::= character ∣ number [sequence]; answer ::= block ∣ block answer;

\begin{aligned} &\text{character} ::= \text{a} \mid \text{b} \mid \text{c}; \\ &\text{sequence} ::= \text{character} \mid \text{character} \ \text{sequence}; \\ &\text{block} ::= \text{character} \mid \text{number} [\ \text{sequence} \ ]; \\ &\text{answer} ::= \text{block} \mid \text{block} \ \text{answer}; \end{aligned}

Here, number ( $1 \leq$ number $< 1 0^{10}$ < 10^{10}) is a positive integer written without leading zeros.

Note that a valid string according to this grammar must not contain any spaces.

The output string must encode a sequence of presses in the optimal solution. The block character corresponds to pressing the respective button, while the block number [ sequence ] corresponds to repeating the sequence of presses number times. See the sample test cases for further clarification.

Examples

Example

1 0 0
0

Example

1 2 2
2
ab

Example

5 1 5
-1

Example 1

Note

In the first example, the lock is already open, so no button presses are needed.

In the second example, the sequence "ab" (press "a", then "b") opens the lock.

In the third example, it is impossible to open the lock.