For problem statement at 1000-1999/1800-1899/1800-1809/1801/problemF.txt this is a correct solution, but verifier at 1000-1999/1800-1899/1800-1809/1801/verifierF.go ends with case 1 failed: expected 0.8750000000 got 0.875000000000
input:
3 7
8 1 8
exit status 1 can you fix the verifier? ```go
package main

import (
	"bufio"
	"container/heap"
	"fmt"
	"math"
	"os"
)

type Option struct {
	lnT float64
	lnU float64
}

type Upgrade struct {
	delta  float64
	i      int
	next   int
	index  int
}
type MinHeap []*Upgrade

func (h MinHeap) Len() int            { return len(h) }
func (h MinHeap) Less(i, j int) bool  { return h[i].delta < h[j].delta }
func (h MinHeap) Swap(i, j int)       { h[i], h[j] = h[j], h[i]; h[i].index = i; h[j].index = j }
func (h *MinHeap) Push(x interface{}) { *h = append(*h, x.(*Upgrade)) }
func (h *MinHeap) Pop() interface{} {
	old := *h
	n := len(old)
	it := old[n-1]
	*h = old[:n-1]
	return it
}

func genOptions(a int64) []Option {
	opts := make([]Option, 0, 2*int(math.Sqrt(float64(a)))+5)
	for l := int64(1); l <= a; {
		t := a / l
		r := a / t
		opts = append(opts, Option{
			lnT: math.Log(float64(t)),
			lnU: math.Log(float64(r)),
		})
		l = r + 1
	}
	return opts
}

func main() {
	in := bufio.NewReader(os.Stdin)
	var n int
	var k int64
	if _, err := fmt.Fscan(in, &n, &k); err != nil {
		return
	}
	a := make([]int64, n)
	for i := 0; i < n; i++ {
		fmt.Fscan(in, &a[i])
	}

	if k <= 0 {
		fmt.Printf("0\n")
		return
	}

	opts := make([][]Option, n)
	sumLnA := 0.0
	for i := 0; i < n; i++ {
		opts[i] = genOptions(a[i])
		sumLnA += math.Log(float64(a[i]))
	}

	lnK := math.Log(float64(k))
	if sumLnA+1e-15 < lnK {
		fmt.Printf("0\n")
		return
	}

	// Binary search on lambda
	l, r := 0.0, 100.0
	for it := 0; it < 80; it++ {
		mid := (l + r) / 2
		sumU := 0.0
		for i := 0; i < n; i++ {
			best := -1e300
			bestU := 0.0
			oi := opts[i]
			for j := 0; j < len(oi); j++ {
				val := oi[j].lnT + mid*oi[j].lnU
				if val > best {
					best = val
					bestU = oi[j].lnU
				}
			}
			sumU += bestU
		}
		if sumU >= lnK {
			r = mid
		} else {
			l = mid
		}
	}

	// Tie-aware selection at lambda = r
	eps := 1e-12
	sumU := 0.0
	sumT := 0.0
	curIdx := make([]int, n)
	cands := make([][]int, n)
	for i := 0; i < n; i++ {
		oi := opts[i]
		best := -1e300
		for j := 0; j < len(oi); j++ {
			val := oi[j].lnT + r*oi[j].lnU
			if val > best {
				best = val
			}
		}
		// collect near-max candidates
		tmp := make([]int, 0, 3)
		for j := 0; j < len(oi); j++ {
			val := oi[j].lnT + r*oi[j].lnU
			if val >= best-eps {
				tmp = append(tmp, j)
			}
		}
		// sort tmp by lnU ascending (small list, insertion sort)
		for x := 1; x < len(tmp); x++ {
			kx := tmp[x]
			y := x - 1
			for y >= 0 && oi[tmp[y]].lnU > oi[kx].lnU {
				tmp[y+1] = tmp[y]
				y--
			}
			tmp[y+1] = kx
		}
		cands[i] = tmp
		curIdx[i] = 0
		sumU += oi[tmp[0]].lnU
		sumT += oi[tmp[0]].lnT
	}

	if sumU+1e-15 < lnK {
		h := &MinHeap{}
		heap.Init(h)
		for i := 0; i < n; i++ {
			if len(cands[i]) >= 2 {
				oi := opts[i]
				j0 := cands[i][0]
				j1 := cands[i][1]
				heap.Push(h, &Upgrade{
					delta: oi[j1].lnU - oi[j0].lnU,
					i:     i,
					next:  1,
				})
			}
		}
		for sumU+1e-15 < lnK && h.Len() > 0 {
			u := heap.Pop(h).(*Upgrade)
			i := u.i
			prev := cands[i][curIdx[i]]
			curIdx[i] = u.next
			now := cands[i][curIdx[i]]
			sumU += opts[i][now].lnU - opts[i][prev].lnU
			sumT += opts[i][now].lnT - opts[i][prev].lnT
			if curIdx[i]+1 < len(cands[i]) {
				jc := cands[i][curIdx[i]]
				jn := cands[i][curIdx[i]+1]
				heap.Push(h, &Upgrade{
					delta: opts[i][jn].lnU - opts[i][jc].lnU,
					i:     i,
					next:  curIdx[i] + 1,
				})
			}
		}
		// If still not enough due to numerical issues, slightly increase lambda and recompute greedily
		if sumU+1e-12 < lnK {
			rr := r + 1e-9
			sumU = 0.0
			sumT = 0.0
			for i := 0; i < n; i++ {
				oi := opts[i]
				best := -1e300
				bestU := 0.0
				bestT := 0.0
				for j := 0; j < len(oi); j++ {
					val := oi[j].lnT + rr*oi[j].lnU
					if val > best {
						best = val
						bestU = oi[j].lnU
						bestT = oi[j].lnT
					}
				}
				sumU += bestU
				sumT += bestT
			}
		}
	}

	lnRes := math.Log(float64(k)) + sumT - sumLnA
	res := math.Exp(lnRes)
	fmt.Printf("%.12f\n", res)
}
```