
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 $\frac 14$-array is a structured, infinite, collection of numbers. For example, a $\frac 14$-array is constructed as follows:
\begin{align*}
1 \qquad \frac 14 \qquad \frac 1{16} \qquad \frac 1{64} \qquad &\cdots \\
\frac 18 \qquad \frac 1{32} \qquad \frac{1}{128} \qquad &\cdots \\
\frac 1{64} \qquad \frac 1{256} \qquad &\cdots \\
\frac 1{512} \qquad &\cdots \\
&\ddots
\end{align*}
In this array, the first entry of each row is $\frac{1}{2p}$ times the first entry of the previous row, where $p=4$. Then, each succeeding term in a row is $\frac{1}{p}$ times the previous term in the same row. Find the sum of all terms in the $\frac{1}{4}$-array, expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers.

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