
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
The sum of 3 consecutive odd integers (the smallest being \( k \)) is 51. Find \( k \).

If \( x^{2}+6x+k \equiv (x+a)^{2}+C \), where \( a \) and \( C \) are constants, find \( C \).

If \(\frac{p}{q} = \frac{q}{r} = \frac{r}{s} = 2\) and \( R = \frac{p}{s} \), find \( R \).

If \( A = \frac{3^{n} \cdot 9^{n+1}}{27^{n-1}} \), find \( A \).

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