
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
$n^{2} (n \geq 4)$ positive numbers are arranged in $n$ rows and $n$ columns,
\begin{tabular}{llllll}
$a_{11}$ & $a_{12}$ & $a_{13}$ & $a_{14}$ & $\cdots$ & $a_{1n}$ \\
$a_{21}$ & $a_{22}$ & $a_{23}$ & $a_{24}$ & $\cdots$ & $a_{2n}$ \\
$a_{31}$ & $a_{32}$ & $a_{33}$ & $a_{34}$ & $\cdots$ & $a_{3n}$ \\
$a_{41}$ & $a_{42}$ & $a_{43}$ & $a_{41}$ & $\cdots$ & $a_{4n}$ \\
$\cdots \cdots$ & & & & & \\
$a_{n1}$ & $a_{n2}$ & $a_{n3}$ & $a_{n4}$ & $\cdots$ & $a_{nn}$
\end{tabular}

where each row is an arithmetic sequence, each column is a geometric sequence, and all common ratios are equal. Given $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, find
$$
a_{11}+a_{22}+a_{33}+a_{44}+\cdots+a_{nn}
$$

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