
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
Two circles with radii 4 and 3 are given on a plane with centers at points \( O_1 \) and \( O_2 \), touching a certain line at points \( M_1 \) and \( M_2 \) and lying on opposite sides of this line. The ratio of the segment \( O_1O_2 \) to the segment \( M_1M_2 \) is equal to \(\frac{2}{\sqrt{3}}\). Find \( O_1O_2 \).

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