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When drawing the path, indicate direction of travel on each segment to help solve this.
While the rules as written will allow you to backtrack - pick up a parchment and leave by the same path you entered on - this then allows multiple solutions, some of which are not the "right" one.
Some paths can be drawn in immediately; e.g. between the U and k and s, since there is only one path into and out of the k. Note that you can also tell the direction of travel due to the lower-case to lower-case rule.
The upper left corner has to be R-->v-->o because of the v having only one entrance and exit..
The lower right corner has to be M-->z-->h-->G-->b-->Z because of the h having only one entrance and exit..