Minimum Steps on a Hexagonal Honeycomb

The honeycomb is formed by a tiling of regular hexagons. We define six directions on the honeycomb as follows:

• 0: West
• 1: West but 60° toward North
• 2: East but 60° toward North
• 3: East
• 4: East but 60° toward South
• 5: West but 60° toward South

For a given origin O, a point A is defined by taking a path starting at O that first moves p steps in direction d and then moves q steps in the direction (d+2) mod 6 (which is equivalent to a clockwise turn of 120° from d). The coordinate of point A is represented as (d, p, q). Note that a single point in the honeycomb might have multiple coordinate representations.

For example, the diagram shows the points B(0, 5, 3) and C(2, 3, 2).

Your task is: given two points represented as (d₁, p₁, q₁) and (d₂, p₂, q₂), determine the minimum number of steps required to travel from one point to the other on the honeycomb grid.

Note on Implementation: To solve this problem, one approach is to convert the given coordinate representation into axial coordinates. In our solution, we use the following mapping for the two basis directions:

• For a coordinate (d, p, q), let the first directional movement be along v[d] and the second one along v[(d+2) mod 6].
• We define the vectors in axial coordinates as follows:
  v[0] = (-1, 0)   (West)
  v[1] = (-1, -1)   (Northwest)
  v[2] = (0, -1)   (Northeast)
  v[3] = (1, 0)   (East)
  v[4] = (1, 1)   (Southeast)
  v[5] = (0, 1)   (Southwest)

The position of the point is computed as:

[ \text{position} = p \cdot v[d] + q \cdot v[(d+2) \bmod 6]. ]

The distance between two points in axial coordinates (x₁, y₁) and (x₂, y₂) can be calculated using:

[ \text{distance} = \frac{|x₁ - x₂| + |y₁ - y₂| + |(x₁+y₁) - (x₂+y₂)|}{2}. ]

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inputFormat

The input consists of two lines. The first line contains three integers: d₁, p₁, q₁ representing the first point. The second line contains three integers: d₂, p₂, q₂ representing the second point.

outputFormat

Output a single integer: the minimum number of steps required to travel from the first point to the second point.

sample

0 5 3
2 3 2

9