
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 $A$ with $AB=3$ and $AC=3\sqrt{3}$. Altitude $AD$ intersects median $BE$ at point $G$. Determine the length of $AG$. [asy]
draw((0,0)--(15,0)--(0,15*sqrt(3))--cycle);
draw((0,0)--(11.25,6.495)); draw((0,15*sqrt(3))--(7.5,0));
draw((10.02,5.79)--(10.725,4.515)--(11.985,5.235));
label("$A$",(0,0),SW); label("$E$",(7.5,0),S); label("$B$",(15,0),SE);
label("$D$",(11.25,6.495),NE); label("$C$",(0,15*sqrt(3)),N); label("$G$",(6.435,3.705),NW);
[/asy]

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