
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
\( A A_{1} \) is the median of triangle \( A B C \). Point \( C_{1} \) lies on side \( A B \) such that \( A C_{1} : C_{1} B = 1 : 2 \). Segments \( A A_{1} \) and \( C C_{1} \) intersect at point \( M \). 

Find the ratios \( A M : M A_{1} \) and \( C M : M C_{1} \).

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