Problem:

You are given a string containing only lowercase alphabetical characters. Your task is to write a Python function that compresses the string using the following rules:

If a character appears only once, it should be followed by a "1".
If a character appears more than once in a row, it should be followed by the count of consecutive occurrences of that character.
The compressed string should only include lowercase alphabetical characters and the counts of their consecutive occurrences. Your function should return the compressed string.

Example:

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Input: "aaabcccaaabbb"
Output: "a3b1c3a3b3"
In this example, the input string has three consecutive 'a's, followed by a single 'b', three consecutive 'c's, three consecutive 'a's, and finally, three consecutive 'b's. The compressed string reflects these counts as "a3b1c3a3b3".