How can I find the original message of two encrypted (Vernam Cipher) text without the key Can I get the steps or the solution if possible?
You know that the two ciphertext you are given are formed as
$$C_1 = S_1 \oplus R\\C_2 = S_2 \oplus R$$ with the same $R$ in both cases.
So you can get rid of $R$ by calculating $$C^\ast = C_1 \oplus C_2 = S_1 \oplus S_2$$
Then you know that there are only two words, $111$ and $0000$. Thus, in $C^\ast$ $1$ can only occur if one sentence contains $A$ while the other contains $B$.
From there, since the two words have different lengths, you should be able to determine candidate sentences.