Kubernetes is a container orchestration system for automating software deployments, and google wants your help to maintain a highly available Kubernetes cluster.

A Kubernetes cluster is a group of $$$N$$$ nodes (machines) that are used to run applications, each node has a capacity of $$$c_i$$$ pods (a pod is an instance of an application).

We have $$$M$$$ applications and each application has a number of pods $$$a_j$$$ that needs to be run on the cluster to make sure that its users have a nice experience, for example a widely used application like google search must have a lot of instances running, while another small application like google keep doesn't need that much instances to keep users happy.

We define for each application a value called $$$availability_j$$$ which is equal to the minimum number of pods from the application $$$j$$$ running on the same node after distrubution for example:

Here we have 3 nodes with capacities $$$5,4,5$$$ respectively and 3 applications:

1. Application 1: 7 pods (1 to 7) with availability of 2 (node 2 is only running 2 pods from this application)
2. Application 2: 5 pods (8 to 12) with availability of 1 (all nodes are running 2 pods from this application except for node 2)
3. Application 3: 1 pod (pod 13) with availabilito of 0 (there is only one pod in node 3 so nodes 1 and 2 have zero pods running from this application)

Now your job is determine the maximum possible value for $$$\min_{1 \le j \le M}(availability_j)$$$

Note: Its guaranteed that all applications can be deployed in the cluster