
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
There are $8 n^{2} (n \geqslant 4)$ positive numbers arranged in rows and columns as follows:
\begin{tabular}{llllll}
$a_{11}$ & $a_{12}$ & $a_{13}$ & $a_{14}$ & $\ldots$ & $a_{1 n}$, \\
$a_{21}$ & $a_{22}$ & $a_{23}$ & $a_{24}$ & $\ldots$ & $a_{2 n}$, \\
$a_{31}$ & $a_{32}$ & $a_{33}$ & $a_{34}$ & $\ldots$ & $a_{3 n}$, \\
$\ldots$ & $\ldots$ & $\ldots$ & $\ldots$ & $\ldots$ & $\ldots$ \\
$a_{n 1}$ & $a_{n 2}$ & $a_{n 3}$ & $a_{n 4}$ & $\ldots$ & $a_{n n}$,
\end{tabular}

In each row, the numbers form an arithmetic sequence. In each column, the numbers form a geometric sequence, and all the common ratios are equal. It is known that $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$. Find $a_{11}+a_{22}+a_{33}+\cdots+a_{n n}$.

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