
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
Given a "triangular array" as shown in the table: $$\left.\begin{matrix}\frac {1}{4} &   &   &   \\ \frac {1}{2} & \frac {1}{4} &   &   \\ \frac {3}{4} & \frac {3}{8} & \frac {3}{16} &   \\ … &   &   &  \end{matrix}\right.$$It is known that the numbers in each column form an arithmetic sequence, and starting from the third row, the numbers in each row form a geometric sequence, with the common ratio of each row being the same. Let the number in the $i$-th row and $j$-th column be $a_{ij}$ ($i \geq j$, $i$, $j \in \mathbb{N}^*$),  
(1) Find $a_{83}$;  
(2) Write the expression of $a_{ij}$ in terms of $i$ and $j$;  
(3) Let the sum of the $n$-th row be $A_n$, find the expression for the sum of the first $m$ terms $B_m$ of the sequence $\{A_n\}$.

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