
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
Given the triangular pyramid $P-ABC$, point $M$ on segment $PC$ satisfies $PM=\frac{1}{3}PC$, and point $N$ on segment $PB$ satisfies $PN=\frac{2}{3}PB$. Calculate the ratio of the volumes of the triangular pyramids $P-AMN$ and $P-ABC$.

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