
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
A solid has a square base of side length $t$. Vertices of the base are $A$, $B$, $C$, and $D$. Two vertices $E$ and $F$ are located above this base such that $AE = t$ and $BF = 2t$, with $EF$ parallel to side $AB$. Extend edges $AE, BF, CE, and DF$ to meet at a point $G$ above the square, thus forming a tetrahedral section atop the square base. Give that $t = 3\sqrt{3}$, calculate the volume of this tetrahedral section.

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