
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 diagram, $\triangle ABC$ is right-angled at $A$ with $AB=3$ and $AC=3\sqrt{3}$. Altitude $AD$ intersects median $CE$ at $G$. Find the length of $AG$.
[asy]
draw((0,0)--(15,0)--(0,15*sqrt(3))--cycle);
draw((0,0)--(11.25,6.5)); draw((0,15*sqrt(3))--(7.5,0));
draw((10.02,5.8)--(10.76,4.515)--(11.93,5.17));
label("$A$",(0,0),SW); label("$E$",(7.5,0),S); label("$B$",(15,0),SE);
label("$D$",(11.25,6.5),NE); label("$C$",(0,15*sqrt(3)),N); label("$G$",(6.44,3.71),NW);
[/asy]

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