
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
Given the tabular information \(3^5=243\), \(3^6=729\), \(2^8=256\), \(2^{10}=1024\), \(10^2=100\), and \(10^3=1000\), the most precise statement one can make about \(\log_{10}{3}\) is between which of the following pairs of fractions?
A) \(\frac{2}{7} \) and \(\frac{1}{2}\)
B) \(\frac{2}{5} \) and \(\frac{1}{2}\)
C) \(\frac{2}{5} \) and \(\frac{3}{7}\)
D) \(\frac{1}{3} \) and \(\frac{1}{2}\)
E) \(\frac{1}{3} \) and \(\frac{3}{7}\)

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