# Practice on Toph

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## ZiGzAg NuMbEr

A number is called a Zig-zag number if the differences between consecutive digits alternate between positive and negative. A Zig-zag number has the following properties.

- It has at least 3 digits.
- Differences between consecutive digits alternate between positive and negative where first difference can be either positive or negative.
- It has no leading 0 (zero).

For example, 174925 is a Zig-zag number because the differences between consecutive digits (6, -3, 5, -7, 3) are alternating between positive and negative. 154, 190, 101, 909 and 13254 are some Zig-zag numbers while 123, 900 and 4556 are not.

You are given two integers **L** and **R**. You need to find all the Zig-zag numbers within the range **L** and **R** inclusive, sort them in ascending order with respect to their Digit Sum and answer the **K ^{th}** Zig-zag number in the sorted sequence.

**Digit Sum** of a number is the sum of all its digits. **Digit Sum** of (40528) = 4+0+5+2+8 = 19.

Let, **x** and **y** are two Zig-zag numbers within the range **L** and **R** inclusive. In sorted sequence of Zig-zag numbers, **x** will appear before **y** if **any** of the followings satisfies:

**Digit Sum of (x) < Digit Sum of (y)****Digit Sum of (x) = Digit Sum of (y)**and**x < y**.

Let, **L** = 120, **R** = 140 and **K** = 6.

From 120 to 140 there are 6 Zig-zag numbers. They are 120, 121, 130, 131, 132 and 140 and their digit sums are 3, 4, 4, 5, 6 and 5 respectively. If we sort them according to the given procedure, we will get the following sequence: 120, 121, 130, 131, 140 and 132. So, 6th Zig-zag number in the sorted sequence is 132.

#### Input

The input begins with a single positive integer **T (≤100000)** on a line by itself indicating the number of the cases. Each of the next **T** lines contains three space separated integers **L** , **R** and **K** **(1 ≤ L ≤ R ≤ 10^18, 1 ≤ K ≤ 10^18)**.

#### Output

For each test case, print the test case number followed by **K ^{th}** Zig-zag number in the sorted sequence. If there is no such Zig-zag number, print

**-1**. You should follow the exact format as the sample output.

#### Samples

Input | Output |
---|---|

3 153 165 3 100 165 30 1 142 11 | Case 1: 153 Case 2: -1 Case 3: 132 |

#### Explanation

**Case 1:** There are 8 Zig-zag numbers from 153 to 165. If we sort them according to the given procedure, we get the following sequence: 160, 161, 153, 162, 154, 163, 164 and 165.

**Case 2:** There are less than 30 Zig-zag numbers in the given range.

#### Warning

The input file is huge, please use faster I/O methods.

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