
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 $G$. Find the length of $AG$. Assume the diagram has this configuration: 

[Contextual ASY Diagram]
```
draw((0,0)--(9,0)--(0,10*sqrt(3))--cycle);
draw((0,0)--(7.5,4.33)); draw((0,10*sqrt(3))--(4.5,0));
draw((6.68,3.86)--(7.17,3.01)--(7.99,3.49));
label("$A$",(0,0),SW); label("$E$",(4.5,0),S); label("$B$",(9,0),SE);
label("$D$",(7.5,4.33),NE); label("$C$",(0,10*sqrt(3)),N); label("$G$",(4.29,2.47),NW);
```

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