
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, under what conditions will the expressions $\sqrt{a+\frac{b}{c}}$ and $a\sqrt{\frac{b}{c}}$ be equal, if $b=pd$ where $p$ and $d$ are positive integers?
A) \( c = apd \)
B) \( c = pd \)
C) \( c = apd - \frac{pd}{a}, \text{ and } (pd \mod a) = 0 \)
D) \( c = p(d - a) \)

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