
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
Positive real numbers $c$ and $d$ satisfy the equation
\[
\sqrt{\log{c}} + \sqrt{\log{d}} + \log \sqrt{c} + \log \sqrt{d} + \log(\sqrt{c} \cdot \sqrt{d}) = 50
\]
where each term on the left is a positive integer and $\log$ denotes the base 10 logarithm. What is $cd$?
A) $10^{35}$  
B) $10^{36}$  
C) $10^{37}$  
D) $10^{38}$  
E) $10^{39}$

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