P1573 Stack Operations

Manager’s note: Since the proof is extremely difficult but the problem itself is easy to solve, this problem is rated as a gray problem.

There are four stacks. The first three are empty, and the fourth stack contains, from top to bottom, $1,2,3,\cdots ,n$. Each stack supports only one operation: pop and push. This means popping the top element $x$ from some stack A and immediately pushing it onto any stack B. However, such an operation is allowed only if the following rules are satisfied. - Rule $1$: Stack A must be non-empty. - Rule $2$: Stack B is empty, or $x$ is smaller than the top of stack B. Given $n$, find the minimal number of operations needed to move all $n$ elements from the fourth stack to the first stack. Since the minimal number of operations can be very large, output the answer modulo $10^6+7$.

One line containing an integer $n$.

One line containing a positive integer, the value of the minimal number of operations $\bmod (10^6+7)$.

- For 30% of the testdata, $n \le 8$. - For 60% of the testdata, $n \le 60$. - For 100% of the testdata, $n \le 2 \times 10^9$. Translated by ChatGPT 5