<p>There is a donuts shop that bakes donuts in batches of <code>batchSize</code>. They have a rule where they must serve <strong>all</strong> of the donuts of a batch before serving any donuts of the next batch. You are given an integer <code>batchSize</code> and an integer array <code>groups</code>, where <code>groups[i]</code> denotes that there is a group of <code>groups[i]</code> customers that will visit the shop. Each customer will get exactly one donut.</p>

<p>When a group visits the shop, all customers of the group must be served before serving any of the following groups. A group will be happy if they all get fresh donuts. That is, the first customer of the group does not receive a donut that was left over from the previous group.</p>

<p>You can freely rearrange the ordering of the groups. Return <em>the <strong>maximum</strong> possible number of happy groups after rearranging the groups.</em></p>

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

<pre>
<strong>Input:</strong> batchSize = 3, groups = [1,2,3,4,5,6]
<strong>Output:</strong> 4
<strong>Explanation:</strong> You can arrange the groups as [6,2,4,5,1,3]. Then the 1<sup>st</sup>, 2<sup>nd</sup>, 4<sup>th</sup>, and 6<sup>th</sup> groups will be happy.
</pre>

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

<pre>
<strong>Input:</strong> batchSize = 4, groups = [1,3,2,5,2,2,1,6]
<strong>Output:</strong> 4
</pre>

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

<ul>
	<li><code>1 &lt;= batchSize &lt;= 9</code></li>
	<li><code>1 &lt;= groups.length &lt;= 30</code></li>
	<li><code>1 &lt;= groups[i] &lt;= 10<sup>9</sup></code></li>
</ul>


class Solution:
    def maxHappyGroups(self, batchSize: int, groups: List[int]) -> int:
        if batchSize == 1:
            return len(groups)
        if batchSize == 2:
            return len(groups) - sum(g % 2 for g in groups) // 2

        klasses = [0] * batchSize
        for g in groups:
            if g % batchSize != 0:
                klasses[g % batchSize] += 1
        res_sm = sum(k_r * r
                     for k_r, r in zip(klasses, range(batchSize)))
        offset = (0, *accumulate(klasses))

        cur_res_sm = 0
        state = [0] * batchSize

        @cache
        def happy(istate: int) -> int:
            nonlocal cur_res_sm
            if cur_res_sm >= res_sm:
                return 0

            r = 0
            for klass in range(1, batchSize):
                if klasses[klass] <= state[klass]:
                    continue

                cur_res_sm += klass
                state[klass] += 1

                r = max(
                    r, happy(istate | (1 << (offset[klass] + state[klass]))))

                cur_res_sm -= klass
                state[klass] -= 1

            return r + int(cur_res_sm % batchSize == 0)

        zero_res_cnt = len(groups) - offset[batchSize]
        return zero_res_cnt + happy(0)

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