
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
Find the valuation...

(i) 3-adic valuation of \( A = 2^{27} + 1 \).

(ii) 7-adic valuation of \( B = 161^{14} - 112^{14} \).

(iii) 2-adic valuation of \( C = 7^{20} + 1 \).

(iv) 2-adic valuation of \( D = 17^{48} - 5^{48} \).

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