For problem statement at 1000-1999/1300-1399/1310-1319/1311/problemE.txt this is a correct solution, but verifier at 1000-1999/1300-1399/1310-1319/1311/verifierE.go ends with test 6 failed
expected:
YES
1 2 2 3 3 5 5 6 6

got:
YES
1 1 2 2 4 4 6 8 9

exit status 1 can you fix the verifier? ```go
package main

import (
	"bufio"
	"fmt"
	"os"
)

func main() {
	in := bufio.NewReader(os.Stdin)
	out := bufio.NewWriter(os.Stdout)
	defer out.Flush()

	var t int
	if _, err := fmt.Fscan(in, &t); err != nil {
		return
	}

	for i := 0; i < t; i++ {
		var n, d int
		fmt.Fscan(in, &n, &d)
		solve(out, n, d)
	}
}

func solve(out *bufio.Writer, n, d int) {
	// Maximum sum of depths is when the tree is a path (0, 1, ..., n-1)
	maxSum := n * (n - 1) / 2
	if d > maxSum {
		fmt.Fprintln(out, "NO")
		return
	}

	// Minimum sum of depths is when the tree is a complete binary tree
	minSum := 0
	rem := n
	width := 1
	depth := 0
	for rem > 0 {
		take := width
		if rem < take {
			take = rem
		}
		minSum += take * depth
		rem -= take
		width *= 2
		depth++
	}

	if d < minSum {
		fmt.Fprintln(out, "NO")
		return
	}

	fmt.Fprintln(out, "YES")

	// Start with a chain: 0-1-2-...-(n-1)
	// cnt[i] is the number of nodes at depth i
	cnt := make([]int, n)
	for i := range cnt {
		cnt[i] = 1
	}

	currentSum := maxSum
	deepest := n - 1 // The deepest level that currently has a node
	target := 1      // The shallowest level that can accept a child

	for currentSum > d {
		// Find the first level 'target' that is not full.
		// A level h is not full if cnt[h] < 2 * cnt[h-1].
		for target < deepest && cnt[target] == 2*cnt[target-1] {
			target++
		}

		// Calculate how much we can reduce the sum by moving a node from 'deepest' to 'target'
		move := deepest - target
		// Calculate how much we NEED to reduce
		diff := currentSum - d

		if move > diff {
			// We need less than the max possible move.
			// Move the node from 'deepest' to 'deepest - diff'
			finalDepth := deepest - diff
			cnt[deepest]--
			cnt[finalDepth]++
			currentSum -= diff
			break
		} else {
			// Move the node from 'deepest' to 'target'
			cnt[deepest]--
			cnt[target]++
			currentSum -= move
			// If we removed the last node from 'deepest', update 'deepest'
			if cnt[deepest] == 0 {
				deepest--
			}
		}
	}

	// Construct parents array from cnt distribution
	parents := make([]int, n+1)
	layer := []int{1} // Vertex IDs at the current level
	nextID := 2

	for h := 1; h < n; h++ {
		if cnt[h] == 0 {
			break
		}
		numNodes := cnt[h]
		newLayer := make([]int, 0, numNodes)
		parentIdx := 0
		degree := 0 // Number of children assigned to current parent

		for i := 0; i < numNodes; i++ {
			p := layer[parentIdx]
			parents[nextID] = p
			newLayer = append(newLayer, nextID)
			nextID++

			degree++
			if degree == 2 {
				degree = 0
				parentIdx++
			}
		}
		layer = newLayer
	}

	// Print parents for vertices 2 to n
	for i := 2; i <= n; i++ {
		if i > 2 {
			fmt.Fprint(out, " ")
		}
		fmt.Fprint(out, parents[i])
	}
	fmt.Fprintln(out)
}
```