
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 triangle \( ABC \), the bisectors \( AA_1 \), \( BB_1 \), \( CC_1 \) are drawn. Let \( L \) be the intersection point of segments \( B_1C_1 \) and \( AA_1 \), \( K \) be the intersection point of segments \( B_1A_1 \) and \( CC_1 \), \( M \) be the intersection point of \( BK \) and \( AA_1 \), and \( N \) be the intersection point of \( BL \) and \( CC_1 \). Find the ratio \( MS: SN \) if \( S \) is the intersection point of the bisector \( BB_1 \) with the segment \( MN \), and \( AB: BC: AC = 2: 3: 4 \). (16 points)

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