
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
In this rule from Aryabhata, a verbal expression for the number of terms in an arithmetic progression is given by the formula:

\[ n=\frac{1}{2}\left[1+\frac{-2 a+\sqrt{(r-2 a)^{2}+8 S r}}{r}\right] \]

where \( n \) is the number of terms, \( a \) is the first term, \( r \) is the common difference, and \( S \) is the sum of all terms. Verify the correctness of this formula.

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