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The Pilots Brothers' refrigerator door has 16 handles, each in one of two states: open or closed. The refrigerator opens only when all handles are open. Changing a handle's state also toggles all handles in its row and column. The challenge is to determine the minimum number of handle switches needed to open the refrigerator and provide a sequence of switches.

The input consists of four lines, each describing the initial state of four handles. Each character represents a handle: '+' means closed, '-' means open. At least one handle is initially closed.

The output should include the minimum number of switches and the sequence of switches required. The sequence can be any valid solution.

For example, consider the following input:

-+-----------+--

The output might be:

61 11 31 44 14 34 4

This solution represents a sequence of switches that leads to all handles being open with the minimum number of operations.

The problem can be approached by modeling the handles as a 4x4 matrix and determining the optimal sequence of row and column toggles to achieve the desired state. The solution leverages the properties of linear algebra over the field GF(2), where each toggle operation is equivalent to adding a vector to the current state.

The minimal number of operations is determined by finding the smallest set of row and column toggles that transforms the initial state to the all-open state. The sequence provided in the output is one possible way to achieve this transformation.

The problem is a classic example of solving a system of linear equations over GF(2), where each equation corresponds to a handle's state. The solution involves finding a combination of row and column operations that results in all handles being open.

The sequence provided is a specific solution that works for the given initial state. The minimal number of operations ensures that the refrigerator door is opened with the fewest possible handle switches.

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