1. **Problem Statement:** Given a positive integer $N$, find the starting number $\leq N$ that produces the longest Collatz sequence chain. If multiple numbers have the same longest chain length, return the maximum such number. 2. **Collatz Sequence Rules:** - If $n$ is even, the next term is $\frac{n}{2}$. - If $n$ is odd, the next term is $3n + 1$. 3. **Chain Length Definition:** The length of the chain is the number of terms from the starting number down to 1, inclusive. 4. **Approach:** - For each number $1 \leq k \leq N$, compute the length of its Collatz sequence. - Use memoization to store chain lengths for numbers already computed to avoid redundant calculations. - Track the number with the longest chain length up to each $N$. 5. **Example:** For $n=9$, the sequence is: $$9 \to 28 \to 14 \to 7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26 \to 13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1$$ which has 20 terms. 6. **Algorithm Steps:** - Initialize an array `lengths` to store chain lengths for numbers up to $5 \times 10^6$. - For each number from 1 to max $N$ in input, compute chain length using recursion and memoization. - Maintain an array `max_num` where `max_num[i]` stores the number $\leq i$ with the longest chain. - For each test case, output `max_num[N]`. 7. **Complexity:** - Preprocessing all chain lengths up to $5 \times 10^6$ is efficient with memoization. - Query answering is $O(1)$ after preprocessing. **Final answer:** For each input $N$, print the number $\leq N$ with the longest Collatz chain length, choosing the maximum number if ties occur.