
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In the interior of triangle $ABC$, point $P$ is marked. Let $A_1, B_1, C_1$ be the intersection points of the line connecting $P$ with $A, B,$ and $C$ with the opposite sides. For which points $P$ is it true that

$$
\frac{A A_{1}}{P A_{1}}+\frac{B B_{1}}{P B_{1}}+\frac{C C_{1}}{P C_{1}} \geqq 9
$$

Present the answer in LaTex format: \boxed{Your answer}
user