
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
[asy] pair A = (0,0), B = (6,4.8), C = (12, 0), D = (3, -6), E = (3, 0), F = (6,0); draw(A--B--C--D--cycle,dot); draw(A--E--F--C,dot); draw(D--E--F--B,dot); markscalefactor = 0.1; draw(rightanglemark(B, A, D)); draw(rightanglemark(D, E, C)); draw(rightanglemark(B, F, A)); draw(rightanglemark(D, C, B)); MP("A",(0,0),W); MP("B",(6,4.8),N); MP("C",(12,0),E); MP("D",(3,-6),S); MP("E",(3,0),N); MP("F",(6,0),S); [/asy]
In the quadrilateral $ABCD$ with right angles at $A$ and $C$, points $E$ and $F$ are on $\overline{AC}$, and $\overline{DE}$ and $\overline{BF}$ are perpendicular to $\overline{AC}$. Given that $AE = 3$, $DE = 6$, and $CE = 9$ determine the length of $BF$.

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