
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

Point \( D \) lies on the side \( A C \) of triangle \( A B C \). The circle \( S_{1} \), inscribed in triangle \( A B D \), touches segment \( B D \) at point \( M \); the circle \( S_{2} \), inscribed in triangle \( B C D \), touches segment \( D C \) at point \( N \). The ratio of the radii of circles \( S_{1} \) and \( S_{2} \) is \( \frac{7}{4} \). It is known that \( B M = 3, M N = N D = 1 \). Find the sides of triangle \( A B C \).

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