Find the minimum height of a rooted tree$$$^{\dagger}$$$ with $$$a+b+c$$$ vertices that satisfies the following conditions:

$$$a$$$ vertices have exactly $$$2$$$ children,
$$$b$$$ vertices have exactly $$$1$$$ child, and
$$$c$$$ vertices have exactly $$$0$$$ children.

If no such tree exists, you should report it. $\text{[math]}$

The tree above is rooted at the top vertex, and each vertex is labeled with the number of children it has. Here $$$a=2$$$, $$$b=1$$$, $$$c=3$$$, and the height is $$$2$$$.

$$$^{\dagger}$$$ A rooted tree is a connected graph without cycles, with a special vertex called the root. In a rooted tree, among any two vertices connected by an edge, one vertex is a parent (the one closer to the root), and the other one is a child.

The distance between two vertices in a tree is the number of edges in the shortest path between them. The height of a rooted tree is the maximum distance from a vertex to the root.

Input

The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases.

The only line of each test case contains three integers $$$a$$$, $$$b$$$, and $$$c$$$ ($$$0 \leq a, b, c \leq 10^5$$$; $$$1 \leq a + b + c$$$).

The sum of $$$a + b + c$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output

For each test case, if no such tree exists, output $$$-1$$$. Otherwise, output one integer — the minimum height of a tree satisfying the conditions in the statement.

Note

The first test case is pictured in the statement. It can be proven that you can't get a height smaller than $$$2$$$.

In the second test case, you can form a tree with a single vertex and no edges. It has height $$$0$$$, which is clearly optimal.

In the third test case, you can form a tree with two vertices joined by a single edge. It has height $$$1$$$, which is clearly optimal.

Idx	Name	Rating
A	Stair, Peak, or Neither?	800
B	Upscaling	800
C	Clock Conversion	800
D	Product of Binary Decimals	1100
E	Nearly Shortest Repeating Substring	1500
F	0, 1, 2, Tree!	1700
G	Shuffling Songs	1900

Codeforces Round 937 (Div. 4)