
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 $ABC$, points $D$ and $E$ are on $\overline{AB}$ and $\overline{AC},$ respectively, and the angle bisector $\overline{AT}$ intersects $\overline{DE}$ at point $F$. Given $AD = 2$, $DB = 2$, $AE = 3$, and $EC = 3$, compute both $\frac{AF}{AT}$ and $\frac{AT}{BT}$.

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