A company of \( n \) soldiers is such that:
1. \( n \) is a palindrome number (reads the same forwards and backwards).
2. When the soldiers arrange in rows of 3, 4, or 5 soldiers, the last row contains 2, 3, and 5 soldiers, respectively.

Find the smallest \( n \) satisfying these conditions and prove that there are infinitely many such numbers \( n \).

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