
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 a plane, there are two circles with radii 12 and 7, centered at points $O_1$ and $O_2$, respectively. They touch a certain line at points $M_1$ and $M_2$ and lie on the same side of this line. The ratio of the length of the segment $M_1 M_2$ to the length of the segment $O_1 O_2$ is $\frac{2 \sqrt{5}}{5}$. Find $M_1 M_2$.

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