The Collatz conjecture is one of the most famous unsolved problems in mathematics. However, the Collatz sequence can also be viewed in a binary fashion depending on whether each value steps up (3x+1) or down (x/2) from the previous value, denoting these steps with "u", "d", respectively. For example, starting from n=12 here is the sequence and step shape:

Depending of the data type, you function must return the starting value (if string shape of letters "u", "d" is given) or shape of steps (if value is given).

This function must also recognize that some shape strings are impossible as entailed by the Collatz transition rules, and correctly return None for all such shapes. You should start from the goal state 1, and perform the given transitions in reverse. Along the way, you have to ensure that your function does not accept moves that would be illegal in the original forward-going Collatz sequence.

Input: String (str) or integer (int).

Output: String (str), integer (int) or None.

Examples:

assert collatz_convert("ududududddddudddd") == 15
assert collatz_convert(135) == "udududduddddududdudduddddudududdudddudddd"
assert collatz_convert("dudududdudddudddd") == 14
assert collatz_convert(10) == "dudddd"

The mission was taken from Python CCPS 109. It is taught for Ryerson Chang School of Continuing Education by Ilkka Kokkarinen