
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 triangle $ABC$, angle $C$ is trisected by lines $CF$ and $CG$, where $F$ and $G$ are points on side $AB$ such that $F$ is closer to $A$ than $G$. Determine the relationship between the lengths $AF$, $FB$, $FG$, and $GB$ if $\frac{AF}{FB} = \frac{CG}{CF}$.
A) $AF = FG = GB = \frac{1}{3}AB$
B) $AF = FG = \frac{1}{2}AB; GB = \frac{1}{6}AB$
C) $AF = \frac{1}{4}AB; FG = \frac{1}{2}AB; GB = \frac{1}{4}AB$
D) $AF = GB = \frac{1}{3}AB; FG = \frac{1}{4}AB$

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