
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 modified $\frac 1p$ -array is constructed as follows with an additional Fibonacci progression:
\begin{align*}
1 \qquad \frac 15\,\ \qquad \frac 1{25}\,\ \qquad \frac{1}{125} \qquad & \cdots \\
\frac{1}{10} \qquad \frac{1}{50} \qquad \frac{1}{250} \qquad & \cdots \\
\frac{1}{100} \qquad \frac{1}{500} \qquad & \cdots \\
\frac{1}{1000} \qquad & \cdots \\
& \ddots
\end{align*}
Where:
- The first entry of each row is $\frac{1}{3p}$ times the first entry of the previous row.
- Each succeeding term in a row is $\frac{1}{p}$ times the previous term in the same row.
- The first element of each row is additionally multiplied by the Fibonacci number corresponding to the row number.

If the sum of all the terms in the modified $\frac{1}{5}$-array can be calculated, express this sum in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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