<p>Given an array of <code>points</code> where <code>points[i] = [x<sub>i</sub>, y<sub>i</sub>]</code> represents a point on the <strong>X-Y</strong> plane, return <em>the maximum number of points that lie on the same straight line</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/02/25/plane1.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> points = [[1,1],[2,2],[3,3]]
<strong>Output:</strong> 3
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/02/25/plane2.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> points = [[1,1],[3,2],[5,3],[4,1],[2,3],[1,4]]
<strong>Output:</strong> 4
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= points.length &lt;= 300</code></li>
	<li><code>points[i].length == 2</code></li>
	<li><code>-10<sup>4</sup> &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 10<sup>4</sup></code></li>
	<li>All the <code>points</code> are <strong>unique</strong>.</li>
</ul>

import collections
class Solution:
    def maxPoints(self, points: List[List[int]]) -> int:
      points = sorted(points, key=lambda v:v[0] * 10001 + v[1])
      if len(points) < 2:
        return len(points)
      primes = list(range(201))
      for i in range(2, len(primes)):
        p = primes[i]
        if not p:
          continue
        for j in range(2*p, len(primes), p):
          primes[j] = 0
      primes = [v for v in primes if v > 1]

      N = len(points)
      slopes = collections.defaultdict(set)
      for s in range(N):
        xs, ys = points[s]
        for e in range(s + 1, N):
          xe, ye = points[e]
          lx, ly = xe - xs, ye - ys
          if lx == 0:
            ly = abs(ly)
          if ly == 0:
            lx = abs(lx)
          for p in primes:
            while lx % p == 0 and ly % p == 0:
              lx, ly = lx // p, ly // p
          key = (lx, ly, xs, ys)
          slopes[key] |= {s, e}
      return max(len(v) for v in slopes.values())
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