
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]


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For a point  $O$  inside a triangle  $ABC$ , denote by  $A_1,B_1, C_1,$  the respective intersection points of  $AO, BO, CO$  with the corresponding sides. Let 
\[n_1 =\frac{AO}{A_1O}, n_2 = \frac{BO}{B_1O}, n_3 = \frac{CO}{C_1O}.\] 
What possible values of  $n_1, n_2, n_3$  can all be positive integers?

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