1950F · 0, 1, 2, Tree!

Problem: 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.

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 Format: 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 Format: 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.