
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 the diagram, $\triangle ABC$ is right-angled at $A$ and has $AB=6$ and $AC=6$.  Altitude $AD$ intersects median $BE$ at point $F$.  What is the length of $AF$?
[asy]
draw((0,0)--(8,0)--(0,8)--cycle);
draw((0,0)--(4,4)); draw((0,8)--(4,0));
draw((3,3)--(3.5,2.25)--(4.5,2.75));
label("$A$",(0,0),SW); label("$E$",(4,0),S); label("$C$",(8,0),SE);
label("$D$",(4,4),NE); label("$B$",(0,8),N); label("$F$",(3,3),NW);
[/asy]

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