Alice and Bob created $N$ and $M$ recipes, respectively ($N, M \ge 1$), and submitted them to Chef for evaluation. Each recipe is represented by a string containing only lowercase English letters. Let's denote Alice's recipes by $A_1, A_2, \ldots, A_N$ and Bob's recipes by $B_1, B_2, \ldots, B_M$.
Accidentally, Chef mixed up those recipes ― now, he has $L = N+M$ recipes in a sequence $S_1, S_2, \ldots, S_L$. Thankfully, the recipes created by Alice and Bob are distinguishable from each other. It is well-known that for each recipe $s$ created by Alice, the following property holds, and for each recipe created by Bob, it does not hold: For each $1 \le l < r \le |s|$, the substring $s_l, s_{l+1}, \ldots, s_r$ contains at least as many vowels as consonants. The letters 'a', 'e', 'i', 'o', 'u' are vowels, while the other letters are consonants.
The score of a candidate who made $K$ recipes is calculated as the product of $\frac{x_c}{fx_c^K}$ for all letters $c$ that occur in at least one of these recipes; here, $x_c$ is the number of recipes which contain the letter $c$ and $fx_c$ is the total number of occurrences of this letter in all $K$ recipes.
Let's denote the scores of Alice and Bob by $sc_A$ and $sc_B$ respectively. Chef wants to know their ratio $sc_A/sc_B$. We know that Chef is a legendary cook, but he is not very good at calculating, so he is asking you to find that number.

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first line of each test case contains a single integer $L$.
- $L$ lines follow. For each valid $i$, the $i$-th of these lines contains a single string $S_i$.

-----Output-----
For each test case, if the ratio of scores exceeds $10^7$, print a single line containing the string "Infinity" (without quotes); otherwise, print a single line containing one real number $sc_A/sc_B$.
Your answer will be considered correct if its absolute or relative error does not exceed $10^{-6}$. It is guaranteed that $sc_A/sc_B$ does not lie in the range $10^7 \pm 10$.

-----Constraints-----
- $1 \le T \le 10^5$
- $2 \le L \le 10^5$
- $2 \le |S_i| \le 10^5$ for each valid $i$
- for each valid $i$, $S_i$ contains only lowercase English letters
- the sum of $|S_1| + |S_2| + \ldots + |S_L|$ over all test cases does not exceed $10^7$

-----Subtasks-----
Subtask #1 (25 points):
- $L \le 10$
- $|S_i| \le 10$ for each valid $i$
Subtask #2 (75 points): original constraints

-----Example Input-----
2
4
aba
abc
bab
aac
3
aba
baab
abc

-----Example Output-----
1.1250000
0.0277778

-----Explanation-----
Example case 1: The recipes "aba" and "aac" are created by Alice, while the recipes "abc" and "bab" are created by Bob. The scores are:
- $sc_A = \frac{x_a}{fx_a^N} \cdot \frac{x_b}{fx_b^N} \cdot \frac{x_c}{fx_c^N} = \frac{2}{4^2} \cdot \frac{1}{1^2} \cdot \frac{1}{1^2} = \frac{1}{8}$
- $sc_B = \frac{x_a}{fx_a^M} \cdot \frac{x_b}{fx_b^M} \cdot \frac{x_c}{fx_c^M} = \frac{2}{2^2} \cdot \frac{2}{3^2} \cdot \frac{1}{1^2} = \frac{1}{9}$
- $\frac{sc_A}{sc_B} = \frac{1/8}{1/9} = 1.125$

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

ns = {"__name__": "__main__"}
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = '# cook your dish here\nimport math\nimport decimal\ndecimal.getcontext().prec = 1000\n\nt=int(input())\n\nfor _ in range(t):\n\n l=int(input())\n alice=[]\n bob=[]\n alice_all=""\n bob_all=""\n vowels=[\'a\',\'e\',\'i\',\'o\',\'u\']\n for i in range(l):\n  temp=input()\n  consonant_index=-1\n  for j in range(len(temp)):\n   if temp[j] in vowels:\n    continue\n   else:\n    if consonant_index==-1:\n     consonant_index=j\n    else:\n     if j-consonant_index<=2:\n      bob.append(temp)\n      bob_all=bob_all+temp\n      break\n     else:\n      consonant_index=j\n  else:\n   alice.append(temp)\n   alice_all=alice_all+temp\n\n if l<=10:\n  n=len(alice)\n  m=len(bob)\n\n  score_A,score_B=1,1\n  x_a,x_b=1,1\n  fx_a,fx_b=1,1\n  for i in range(97,123):\n   char=chr(i)\n   if char in alice_all:\n    rec_a=[x for x in alice if char in x]\n    x_a*=len(rec_a)\n    fx_a=fx_a * alice_all.count(char)\n    \n\n   if char in bob_all:\n    rec_b=[x for x in bob if char in x]\n    x_b*=len(rec_b)\n    fx_b=fx_b * bob_all.count(char)\n\n  fx_a=math.pow(fx_a,n)\n  fx_b=math.pow(fx_b,m) \n  score_A=x_a/fx_a\n  score_B=x_b/fx_b\n  ans=score_A/score_B\n  if ans>10000000:\n   print("Infinity")\n  else:\n   print("{0:.9f}".format(ans))\n else:\n\n  n=decimal.Decimal(len(alice))\n  m=decimal.Decimal(len(bob))\n\n  score_A=decimal.Decimal(1).quantize(decimal.Decimal(\'0.0000000001\'))\n  score_B=decimal.Decimal(1).quantize(decimal.Decimal(\'0.0000000001\'))\n  x_a=decimal.Decimal(1).quantize(decimal.Decimal(\'0.0000000001\'))\n  x_b=decimal.Decimal(1).quantize(decimal.Decimal(\'0.0000000001\'))\n  denom_a=decimal.Decimal(1).quantize(decimal.Decimal(\'0.0000000001\'))\n  denom_b=decimal.Decimal(1).quantize(decimal.Decimal(\'0.0000000001\'))\n  fx_a=decimal.Decimal(1).quantize(decimal.Decimal(\'0.0000000001\'))\n  fx_b=decimal.Decimal(1).quantize(decimal.Decimal(\'0.0000000001\'))\n  ans=decimal.Decimal(1).quantize(decimal.Decimal(\'0.0000000001\'))\n  charInA=\'\'.join(sorted(set(alice_all)))\n  charInB=\'\'.join(sorted(set(bob_all)))\n  ctx=decimal.getcontext().copy()\n  ctx.Emax=decimal.MAX_EMAX\n  ctx.Emin=decimal.MIN_EMIN\n  ctx.prec=100\n\n\n  for i in charInA:\n   rec_a=[x for x in alice if i in x]\n   if len(rec_a)>0:\n    x_a=ctx.multiply(x_a,len(rec_a))\n    fx_a=ctx.multiply(fx_a,alice_all.count(i))\n\n  for i in charInB:\n   rec_b=[x for x in bob if i in x]\n   if len(rec_b)>0:\n    x_b=ctx.multiply(x_b,len(rec_b))\n    fx_b=ctx.multiply(fx_b,bob_all.count(i))\n  \n  denom_a=ctx.power(fx_a,n)\n  denom_b=ctx.power(fx_b,m)\n  score_A=ctx.divide(x_a,denom_a)\n  score_B=ctx.divide(x_b,denom_b)\n  \n  ans=ctx.divide(score_A,score_B)\n\n  if ans>10000000:\n   print("Infinity")\n  else:\n   print("{0:.9f}".format(ans))'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 16263