
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 are externally tangent to each other at point \( A \), and both are tangent to a third circle at points \( B \) and \( C \). The extension of chord \( AB \) of the first circle intersects the second circle at point \( D \), and the extension of chord \( AC \) intersects the first circle at point \( E \). The extensions of chords \( BE \) and \( CD \) intersect the third circle at points \( F \) and \( G \) respectively. Find \( BG \) if \( BC = 5 \) and \( BF = 12 \).

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