There are $5$ accents in French, each applicable to only specific letters as follows:

[list]
[*] The cédille: ç
[*] The accent aigu: é
[*] The accent circonflexe: â, ê, î, ô, û
[*] The accent grave: à, è, ù
[*] The accent tréma: ë, ö, ü
[/list]

Cédric needs to write down a phrase in French. He knows that there are $3$ words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If $n$ is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of $n$.
1. **Determine the number of ways to split the letters into 3 words:**
   - We have 12 letters: "cesontoiseaux".
   - We need to place 2 dividers among these 12 letters to create 3 words.
   - The number of ways to place 2 dividers in 12 positions is given by the binomial coefficient:
     \[
     \binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66
     \]

2. **Determine the number of ways to assign accents to each letter:**
   - The letter 'c' can have 2 options (with or without cédille): \(2\) options.
   - The letter 'e' appears twice and can have 5 options each (with or without accent aigu, accent circonflexe, accent grave, or accent tréma): \(5^2 = 25\) options.
   - The letter 's' appears twice and has no accents: \(1\) option.
   - The letter 'o' appears twice and can have 3 options each (with or without accent circonflexe or accent tréma): \(3^2 = 9\) options.
   - The letter 'n' has no accents: \(1\) option.
   - The letter 't' has no accents: \(1\) option.
   - The letter 'i' can have 2 options (with or without accent circonflexe): \(2\) options.
   - The letter 'a' can have 3 options (with or without accent circonflexe or accent grave): \(3\) options.
   - The letter 'x' has no accents: \(1\) option.
   - The letter 'u' can have 4 options (with or without accent circonflexe, accent grave, or accent tréma): \(4\) options.

3. **Calculate the total number of possible phrases:**
   - Multiply the number of ways to split the letters by the number of ways to assign accents:
     \[
     n = 66 \times 2 \times 25 \times 1 \times 9 \times 1 \times 1 \times 2 \times 3 \times 1 \times 4
     \]
   - Simplify the expression:
     \[
     n = 66 \times 2 \times 25 \times 9 \times 2 \times 3 \times 4
     \]

4. **Determine the prime factorization of \( n \):**
   - First, factorize each component:
     \[
     66 = 2 \times 3 \times 11
     \]
     \[
     2 = 2
     \]
     \[
     25 = 5^2
     \]
     \[
     9 = 3^2
     \]
     \[
     2 = 2
     \]
     \[
     3 = 3
     \]
     \[
     4 = 2^2
     \]
   - Combine all the factors:
     \[
     n = 2 \times 3 \times 11 \times 2 \times 5^2 \times 3^2 \times 2 \times 3 \times 2^2
     \]
   - Simplify the combined factors:
     \[
     n = 2^6 \times 3^4 \times 5^2 \times 11
     \]

5. **Count the distinct prime factors:**
   - The distinct prime factors are \(2, 3, 5,\) and \(11\).

Conclusion:
\[
\boxed{4}
\]