
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
Consider a $\frac{1}{9}$ -array constructed as follows:
\begin{align*}
1 \qquad \frac{1}{9} \qquad \frac{1}{81} \qquad \frac{1}{729} \qquad &\cdots\\
\frac{1}{12} \qquad \frac{1}{108} \qquad \frac{1}{972} \qquad &\cdots\\
\frac{1}{144} \qquad \frac{1}{1296} \qquad &\cdots\\
\frac{1}{1728} \qquad &\cdots\\
&\ddots 
\end{align*}
where the first entry of each row is $\frac{1}{4}$ times the first entry of the previous row. Each succeeding term in a row is $\frac{1}{9}$ times the previous term in the same row. If the sum of all the terms in this array can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by $9$.

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