
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
An equilateral triangle \(ABC\) made of plywood is placed on the floor. Three nails are driven into the floor (one close to each side of the triangle) such that the triangle cannot be rotated without lifting it off the floor. The first nail divides side \(AB\) in the ratio \(1:3\) from vertex \(A\), the second nail divides side \(BC\) in the ratio \(2:1\) from vertex \(B\). In what ratio does the third nail divide side \(AC\)?

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