
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 a tetrahedron \( S-ABC \), point \( A_1 \) is the centroid of \( \triangle SBC \). Point \( G \) is on the segment \( AA_1 \) such that \(\frac{|AG|}{|GA_1|}=3\). Line \( SG \) intersects the plane of \( \triangle ABC \) at point \( M \). Determine \(\frac{|A_1 M|}{|AS|} = \quad \).

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