class queue:

  def __init__(self, size):
    self.origsize = size
    self.clear()

  def clear(self):
    self.size = self.origsize
    self.array = [ None ] * self.size
    self.num = 0
    self.begin = 0
    self.end = 0

  def is_empty(self):
    return self.num == 0

  def enqueue(self, item):
    if self.num >= self.size:
      self.grow()
    self.array[self.end] = item
    self.num += 1
    self.end += 1
    if self.end >= self.size:
      self.end = 0

  def dequeue(self):
    if self.num == 0:
      return None
    value = self.array[self.begin]
    self.num -= 1
    self.begin += 1
    if self.begin >= self.size:
      self.begin = 0
    return value

  def grow(self):
    newsize = self.size * 2
    newarray = [ None ] * newsize
    i = self.begin
    j = 0
    n = self.num
    while n > 0:
      item = self.array[i]
      i += 1
      if i >= self.size:
        i = 0
      newarray[j] = item
      j += 1
      n -= 1
    self.size = newsize
    self.array = newarray
    self.begin = 0
    self.end = j

import math

def check(claim, target):
  curr = 4
  print(curr, end = ' ')
  for op in claim:
    if op == "sqrt":
      curr = math.sqrt(curr)
    elif op == "floor":
      curr = math.floor(curr)
    elif op == "fact":
      curr = math.factorial(curr)
    print(curr, end = ' ')
  print()
  return curr == target

def solve(target):
  if target == 4:
    print([ ])
    return
  q = queue(64)
  q.enqueue((4, []))
  while not q.is_empty():
    (curr, path) = q.dequeue()
    if curr == 1:
      continue
    if curr < 1e100:
      newval = math.sqrt(curr)
      newpath = path + [ "sqrt" ]
      if newval == target:
        break
      q.enqueue((newval, newpath))
    if type(curr) == float:
      newval = math.floor(curr)
      newpath = path + [ "floor" ]
      if newval == target:
        break
      q.enqueue((newval, newpath))
    if type(curr) == int and curr < 1000:
      newval = math.factorial(curr)
      newpath = path + [ "fact" ]
      if newval == target:
        break
      q.enqueue((newval, newpath))
  print(newpath)
  if check(newpath, target):
    print("Solution verified")
  else:
    print("Solution is not correct")