A palindrome is a word that reads the same backward as forward. Examples include OMO, lepel, and parterretrap. Determine how many palindromes can be constructed using the letters $a, b, c, d,$ and $e$ under the following conditions:

- Each letter may appear no more than twice in any palindrome.
- The length of each palindrome must be at least $3$ letters.

Note: Any possible combination of letters is considered a word.

Let's think step by step and output the final answer within \boxed{}.
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