
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
Solve for \( k \) given the following condition: \( 3(\log_b x)^2 + 5(\log_c x)^2 = \frac{12(\log x)^2}{\log b \log c} \), where \( b = x^k \) and \( c = x^{1/k} \).
A) \( \sqrt{\frac{6 + \sqrt{21}}{5}} \)
B) \( \sqrt{\frac{6 - \sqrt{21}}{5}} \)
C) \( \frac{\sqrt{21}}{5} \)
D) \( \frac{6}{5} \)

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