
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 $a = 6$ and $b = 3$. Which of the following inequalities correctly orders $\frac{a+3b}{4}, \sqrt[3]{a^2b},$ and $\frac{(a+3b)^2}{4(a+b)}$?
A) $\frac{a+3b}{4} < \sqrt[3]{a^2b} < \frac{(a+3b)^2}{4(a+b)}$
B) $\sqrt[3]{a^2b} < \frac{(a+3b)^2}{4(a+b)} < \frac{a+3b}{4}$
C) $\frac{(a+3b)^2}{4(a+b)} < \frac{a+3b}{4} < \sqrt[3]{a^2b}$
D) $\frac{(a+3b)^2}{4(a+b)} < \sqrt[3]{a^2b} < \frac{a+3b}{4}$
E) $\sqrt[3]{a^2b} < \frac{a+3b}{4} < \frac{(a+3b)^2}{4(a+b)}$

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