
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
Suppose that \( a \) is a positive integer with \( a > 1 \). Determine a closed form expression, in terms of \( a \), equal to

\[ 
1 + \frac{3}{a} + \frac{5}{a^2} + \frac{7}{a^3} + \cdots 
\]

(The infinite sum includes exactly the fractions of the form \(\frac{2k-1}{a^{k-1}}\) for each positive integer \( k \).)

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