
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
Let \( G \) be the centroid of tetrahedron \( A_1 A_2 A_3 A_4 \). Then,
$$
\begin{aligned}
4 G A_1^2 + A_2 A_3^2 + A_2 A_4^2 + A_3 A_4^2 &= 4 G A_2^2 + A_1 A_3^2 + A_1 A_4^2 + A_3 A_4^2 \\
&= 4 G A_3^2 + A_1 A_2^2 + A_1 A_4^2 + A_2 A_4^2 \\
&= 4 G A_4^2 + A_1 A_2^2 + A_1 A_3^2 + A_2 A_3^2 \\
&= \frac{3}{4} \sum_{1 \leq k < j \leq 4} A_k A_j^2.
\end{aligned}
$$

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