Minimize Containers for Coagulation Factors

You are given a multiset of n positive integers \(a_1, a_2, \ldots, a_n\). The task is to partition these numbers into some disjoint sets \(S_1, S_2, \ldots, S_m\) such that each set satisfies one of the following conditions:

1. All numbers in the set are equal. In other words, for every \(a_i, a_j \in S_k\), we have \(a_i = a_j\).
2. All numbers in the set are pairwise distinct. That is, for every \(a_i, a_j \in S_k\) with \(i \neq j\), \(a_i \neq a_j\).

Your goal is to partition the numbers using as few sets (containers) as possible.

Explanation

You can think of the problem as follows: For each type (distinct number) with frequency \(f_i\), you have two ways to place its occurrences:

• Place at most one occurrence in each distinct set. If you decide to use \(K\) distinct sets then at most \(K\) occurrences of any number can be placed in them.
• If the frequency \(f_i\) is greater than \(K\), the remaining \(f_i - K\) copies must be grouped together in a homogeneous container (which counts as one container for that number).

Thus, if you use \(K\) distinct containers, the total number of containers required is:

[ m = K + \Bigl|{ i \mid f_i > K }\Bigr| ]

Your task is to choose an integer \(K\) (with \(0 \le K \le \max_i f_i\)) that minimizes \(m\). Note that choosing \(K = 0\) means you use only homogeneous containers (one per distinct number).

inputFormat

Input is given as follows:

The first line contains a single integer (n) (the number of coagulation factors).

The second line contains (n) positive integers (a_1, a_2, \ldots, a_n) separated by spaces.

outputFormat

Output a single integer representing the minimum number of containers required to partition all coagulation factors.

sample

3
1 2 3

1

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