
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
$\odot O$ and $\odot I$ are the circumcircle and incircle of triangle $ABC$, respectively, with radii 6 and 2. The incircle $\odot I$ is tangent to $AB$, $AC$, and $BC$ at points $F$, $E$, and $D$. $M$ is the centroid of triangle $DEF$. Find the value of $\frac{IM}{OM}$.

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