DeepSeek-R1 and OpenAI o1-Pro Solve an Astrophysics Problem

I gave DeepSeek-R1 and OpenAI o1-Pro a (modified) problem from the book, "Lectures on Astrophysics" by the late Physics Nobel laureate, Steven Weinberg. I was very impressed with both of the models' performance, as each were able to correctly get the final answers to each of the two problems.
Prompt Used:
Solve the following physics problem. Write your solution in LaTeX and enclose your final answer in a box. Problem Statement:
For the following problem, work under the assumption that interstellar matter is in local thermal equilibrium. The ratio of pressure to density is a constant ratio of $v_s^2$ , but the initial density $
ho(r)$ that has the unrealistic form $
ho(r) = \frac{k}{r}$, where $r$ is the distance to a point at $r = 0$ and $k$ is a constant.
1. What is the initial radius of the smallest sphere centered at $r = 0$ that will undergo gravitational collapse?
2. Verify your answer by determining both the kinetic and gravitational self-energy at the value of the radius found in part (a) and verify that the values you find satisfy the viral theorem.