
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
Given $DE = 8$, $EC = 6$, $BC = \frac{1}{3}AD$, and $FB = \frac{2}{3}AD$, find $AG$. Draw your diagram as follows: [asy]
draw((0,0)--(21,0)--(21,14)--cycle);
draw((14,0)--(14,9.333));
draw((7,0)--(7,2.667));
label("A",(0,0),W);
label("B",(7,0),S);
label("G",(7,2.667),N);
label("C",(14,0),S);
label("F",(14,9.333),N);
label("D",(21,0),E);
label("E",(21,14),N);
[/asy]

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