Steve's creeper farm has overflowed and there are creepers everywhere! There are $n$ creepers standing in a line, with the $i$-th creeper having explosive power $e_i$. Steve needs to kill all of them in order to get past.

To do this, he can use his trusty Flint and Steel to detonate creepers. Detonating the creeper at position $i$ kills all creepers at positions $j$ such that $|i - j| \lt e_i$. Dead creepers cannot be detonated. Some creepers may be particularly weak and have explosive power $0$, meaning they cannot be detonated either.

With the Great Hog hot on his tail, there is no time to waste. Find a sequence of detonations that kills all the creepers with as few detonations as possible, or report it is impossible.

Input

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 5 \cdot 10^5$) — the number of creepers.

The second line of each test case contains $n$ integers $e_1, e_2,\ldots, e_n$ ($0 \le e_i \le n$) — the explosive power of each creeper.

It is guaranteed that the sum of $n$ over all test cases does not exceed $5 \cdot 10^5$.

Output

For each test case, output $-1$ if it is impossible to kill all the creepers.

Otherwise, output two lines. On the first line, output a single integer $k$ ($1 \leq k \leq n$), the minimum number of detonations needed. Then output a line with $k$ integers $d_1, d_2, \ldots, d_k$ ($1 \leq d_i \leq n$), stating the sequence of creepers to detonate.

If there are multiple solutions, you can print any of them.

Note

In the first test case, the sequence of detonations is as follows:

• $\underline{0, \mathbf{2}, 2}, 3, 0, 1$
• $\times, \underline{\times, \times, \mathbf{3}, 0, 1}$
• $\times, \times, \times, \times, \times, \times$

where $\times$ represents a dead creeper. Note that if Steve had first detonated creeper $4$ instead, the only creeper left alive would be creeper $1$ which cannot be detonated.

In the second test case, creeper $1$ cannot be detonated, and none of creepers $2$, $3$ or $4$ have a high enough explosive power to kill it, so there is no solution.

In the fifth test case, the sequence of detonations is as follows:

• $\underline{\mathbf{2}, 0}, 2, 4, 2, 2, 4, 1, 1$
• $\times, \times, 2, \underline{4, 2, 2, \mathbf{4}, 1, 1}$
• $\times, \underline{\times, \mathbf{2}, \times}, \times, \times, \times, \times, \times$
• $\times, \times, \times, \times, \times, \times, \times, \times, \times$