<p>There are some red and blue tiles arranged circularly. You are given an array of integers <code>colors</code> and a 2D integers array <code>queries</code>.</p>

<p>The color of tile <code>i</code> is represented by <code>colors[i]</code>:</p>

<ul>
	<li><code>colors[i] == 0</code> means that tile <code>i</code> is <strong>red</strong>.</li>
	<li><code>colors[i] == 1</code> means that tile <code>i</code> is <strong>blue</strong>.</li>
</ul>

<p>An <strong>alternating</strong> group is a contiguous subset of tiles in the circle with <strong>alternating</strong> colors (each tile in the group except the first and last one has a different color from its <b>adjacent</b> tiles in the group).</p>

<p>You have to process queries of two types:</p>

<ul>
	<li><code>queries[i] = [1, size<sub>i</sub>]</code>, determine the count of <strong>alternating</strong> groups with size <code>size<sub>i</sub></code>.</li>
	<li><code>queries[i] = [2, index<sub>i</sub>, color<sub>i</sub>]</code>, change <code>colors[index<sub>i</sub>]</code> to <code>color<font face="monospace"><sub>i</sub></font></code>.</li>
</ul>

<p>Return an array <code>answer</code> containing the results of the queries of the first type <em>in order</em>.</p>

<p><strong>Note</strong> that since <code>colors</code> represents a <strong>circle</strong>, the <strong>first</strong> and the <strong>last</strong> tiles are considered to be next to each other.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">colors = [0,1,1,0,1], queries = [[2,1,0],[1,4]]</span></p>

<p><strong>Output:</strong> <span class="example-io">[2]</span></p>

<p><strong>Explanation:</strong></p>

<p><strong class="example"><img alt="" data-darkreader-inline-bgcolor="" data-darkreader-inline-bgimage="" src="https://assets.leetcode.com/uploads/2024/06/03/screenshot-from-2024-06-03-20-14-44.png" style="width: 150px; height: 150px; padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; --darkreader-inline-bgimage: initial; --darkreader-inline-bgcolor: #181a1b;" /></strong></p>

<p>First query:</p>

<p>Change <code>colors[1]</code> to 0.</p>

<p><img alt="" data-darkreader-inline-bgcolor="" data-darkreader-inline-bgimage="" src="https://assets.leetcode.com/uploads/2024/06/03/screenshot-from-2024-06-03-20-20-25.png" style="width: 150px; height: 150px; padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; --darkreader-inline-bgimage: initial; --darkreader-inline-bgcolor: #181a1b;" /></p>

<p>Second query:</p>

<p>Count of the alternating groups with size 4:</p>

<p><img alt="" data-darkreader-inline-bgcolor="" data-darkreader-inline-bgimage="" src="https://assets.leetcode.com/uploads/2024/06/03/screenshot-from-2024-06-03-20-25-02-2.png" style="width: 150px; height: 150px; padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; --darkreader-inline-bgimage: initial; --darkreader-inline-bgcolor: #181a1b;" /><img alt="" data-darkreader-inline-bgcolor="" data-darkreader-inline-bgimage="" src="https://assets.leetcode.com/uploads/2024/06/03/screenshot-from-2024-06-03-20-24-12.png" style="width: 150px; height: 150px; padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; --darkreader-inline-bgimage: initial; --darkreader-inline-bgcolor: #181a1b;" /></p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">colors = [0,0,1,0,1,1], queries = [[1,3],[2,3,0],[1,5]]</span></p>

<p><strong>Output:</strong> <span class="example-io">[2,0]</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" data-darkreader-inline-bgcolor="" data-darkreader-inline-bgimage="" src="https://assets.leetcode.com/uploads/2024/06/03/screenshot-from-2024-06-03-20-35-50.png" style="width: 150px; height: 150px; padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; --darkreader-inline-bgimage: initial; --darkreader-inline-bgcolor: #181a1b;" /></p>

<p>First query:</p>

<p>Count of the alternating groups with size 3:</p>

<p><img alt="" data-darkreader-inline-bgcolor="" data-darkreader-inline-bgimage="" src="https://assets.leetcode.com/uploads/2024/06/03/screenshot-from-2024-06-03-20-37-13.png" style="width: 150px; height: 150px; padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; --darkreader-inline-bgimage: initial; --darkreader-inline-bgcolor: #181a1b;" /><img alt="" data-darkreader-inline-bgcolor="" data-darkreader-inline-bgimage="" src="https://assets.leetcode.com/uploads/2024/06/03/screenshot-from-2024-06-03-20-36-40.png" style="width: 150px; height: 150px; padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; --darkreader-inline-bgimage: initial; --darkreader-inline-bgcolor: #181a1b;" /></p>

<p>Second query: <code>colors</code> will not change.</p>

<p>Third query: There is no alternating group with size 5.</p>
</div>

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

<ul>
	<li><code>4 &lt;= colors.length &lt;= 5 * 10<sup>4</sup></code></li>
	<li><code>0 &lt;= colors[i] &lt;= 1</code></li>
	<li><code>1 &lt;= queries.length &lt;= 5 * 10<sup>4</sup></code></li>
	<li><code>queries[i][0] == 1</code> or <code>queries[i][0] == 2</code></li>
	<li>For all <code>i</code> that:
	<ul>
		<li><code>queries[i][0] == 1</code>: <code>queries[i].length == 2</code>, <code>3 &lt;= queries[i][1] &lt;= colors.length - 1</code></li>
		<li><code>queries[i][0] == 2</code>: <code>queries[i].length == 3</code>, <code>0 &lt;= queries[i][1] &lt;= colors.length - 1</code>, <code>0 &lt;= queries[i][2] &lt;= 1</code></li>
	</ul>
	</li>
</ul>

def binsearch(l,x):
    lo = -1
    hi = len(l)
    while lo!=hi-1:
        mi = (lo+hi)//2
        if l[mi] < x:
            lo = mi
        elif l[mi] > x:
            hi = mi
        else:
            return mi
    return -1

def find_last_leq(l,x):
    lo = -1
    hi = len(l)
    while lo!=hi-1:
        mi = (lo+hi)//2
        if l[mi] <= x:
            lo = mi
        else:
            hi = mi
    return lo

class Solution:
    def numberOfAlternatingGroups(self, colors: List[int], queries: List[List[int]]) -> List[int]:
        n = len(colors)
        pairs = [] # locations where i == i-1. Notably, areas between adjacent pairs are alternating
        groupsizes = {}
        # if pairs is empty, so should groupsize

        # helpers
        def get_groupsize(p1,p2): # p1 is "before" p2
            gs = p2 - p1
            if gs <= 0:
                gs += n
            return gs
        def addgroup(gs):
            groupsizes[gs] = groupsizes.get(gs,0)+1
        def removegroup(gs):
            groupsizes[gs] -= 1
            if groupsizes[gs] == 0:
                del groupsizes[gs]
        def addpair(index):
            j = find_last_leq(pairs,index)+1
            pairs.insert(j, index)
            if len(pairs)==1: # was empty
                addgroup(n)
            else:
                oldstart = pairs[j-1]
                oldend = pairs[(j+1)%len(pairs)]
                removegroup(get_groupsize(oldstart,oldend))
                addgroup(get_groupsize(oldstart, index))
                addgroup(get_groupsize(index, oldend))
        def removepair(index):
            j = binsearch(pairs,index)
            if j == -1:
                # print("removing nonexistant pair")
                # return
                raise RuntimeError("removing nonexistant pair")
            pairs.pop(j)
            if len(pairs) == 0:
                groupsizes.clear()
            else:
                newstart = pairs[j-1]
                newend = pairs[j%len(pairs)]
                removegroup(get_groupsize(newstart, index))
                removegroup(get_groupsize(index, newend))
                addgroup(get_groupsize(newstart, newend))


        # init
        for i in range(n):
            if colors[i-1] == colors[i]:
                pairs.append(i)
        for j in range(len(pairs)):
            addgroup(get_groupsize(pairs[j-1], pairs[j]))
        # queries
        ans = []
        for q in queries:
            # print(pairs)
            # print(groupsizes)
            # print(q)
            if q[0] == 1:
                tempans = 0
                if len(pairs) == 0:
                    tempans = n
                else:
                    size = q[1]
                    for gs in groupsizes:
                        if gs >= size:
                            tempans += (gs-size+1)*groupsizes[gs]
                ans.append(tempans)
            else:
                index = q[1]
                color = q[2]
                if colors[index] == color:
                    continue
                if colors[index-1] == color and colors[(index+1)%n] == color:
                    # ABA -> AAA
                    addpair(index)
                    addpair((index+1)%n)
                elif colors[index-1] != color and colors[(index+1)%n] != color:
                    # AAA -> ABA
                    removepair(index)
                    removepair((index+1)%n)
                elif colors[index-1] == color and colors[(index+1)%n] != color:
                    # ABB -> AAB
                    addpair(index)
                    removepair((index+1)%n)
                elif colors[index-1] != color and colors[(index+1)%n] == color:
                    # BBA -> BAA
                    removepair(index)
                    addpair((index+1)%n)
                colors[index] = color
        return ans
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