A decrease$-$and$-$conquer algorithm reduces its input size by a rate of $3$ in every iteration $($i$.$e$.,$ from n to n$/3$ to n$/9$ and so on$)$ with a cost of $10$ elementary steps per iteration until the input size is less than $5000.$ At this point the algorithm solves the problem with $25,000,000$ elementary steps. What is the asymptotic time complexity of this algorithm in Big$-$oh notation? Justify your answer.