
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
A $\frac{1}{5}$-array is structured with similar rules:
\[
1 \qquad \frac{1}{5}\quad \frac{1}{25} \quad \frac{1}{125} \quad \cdots\\
\frac{1}{10} \quad \frac{1}{50} \quad \frac{1}{250} \quad \cdots\\
\frac{1}{100} \quad \frac{1}{500} \quad \cdots\\
\frac{1}{1000} \quad \cdots\\
\ddots
\]
Each entry in the $r$th row and the $c$th column is derived from multiplying the first value of the row by $\frac{1}{5}$ repeatedly. The first entry of each row is $\frac{1}{10}$ times the first entry of the previous row. If the sum of all terms in a $\frac{1}{5}$ -array can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, compute $m+n$ modulo $100$.

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