
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
If $a, b, c, d$ are positive integers where $d > 1$, find when the expressions $\sqrt{a^2 + \frac{b^d}{c}}$ and $a^d\sqrt{\frac{b}{c}}$ are equal.
**A)** \( c = \frac{a^d b - b^d}{a^2} \)
**B)** \( c = \frac{a^{2d}b - b^d}{a^2} \)
**C)** \( a = b \text{ and } c \text{ is any value} \)
**D)** \( a = db \text{ and } c = ab \)

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