
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 $a, b$, and $c$ are positive integers such that $a \neq b \neq c$, find the condition under which the expressions $\sqrt{2a+\frac{b}{c}}$ and $2a\sqrt{\frac{b}{c}}$ are equal.
**A) $c = \frac{b(4a - 1)}{2a}$**
**B) $c = \frac{b(3a^2 - 1)}{2a}$**
**C) $c = \frac{b(4a^2 - 2)}{2a}$**
**D) $c = \frac{b(4a^2 - 1)}{2a}$**

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