Die Young

Young diagram is a well known way to describe a partition of a positive integer number. A partition of a number $n$ is a representation as a sum of one or several integer numbers n = m_1 + m_2 + \ldots + m_k where m_1 \geq m_2 \geq \ldots \geq m_k.

A diagram consists of $n$ boxes arranged in $k$ rows, where $k$ is the number of terms in the partition. A row representing the number m_i contains m_i boxes. All rows are left-aligned, and sorted from longest to shortest.

The diagram on the picture below corresponds to the partition 10 = 5 + 3 + 2.

Sometimes it is possible to inscribe one Young diagram into the other. Diagram $X$ can be inscribed into the diagram $Y$ if it is possible to delete some boxes from diagram $Y$ so that it turns to diagram $X$. Note that it is only allowed to remove some boxes, it is not allowed to rotate or flip the diagram. For example, the picture below shows that the diagram for 5 = 3 + 2 can be inscribed into the diagram for 10 = 5 + 3 + 2.

On the other hand, for example, it is impossible to inscribe the diagram for 8 = 4+4 into the diagram for 10 = 5 + 3 + 2.

Given $n$, your task to find such partition of $n$ that the corresponding Young diagram has the greatest possible number of diagrams that can be inscribed into it.

For example, there are $36$ Young diagrams that can be inscribed into the diagram for 10 = 5 + 3 + 2. However, it is not the maximal possible value. The diagram for 10 = 4 + 2 + 2 + 1 + 1 has the better value, there are $41$ diagrams that can be inscribed into it.