
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 regular  $n$ -gon  $A_1A_2\ldots A_n$  with area  $S$ . Let us draw the lines  $l_1,l_2,\ldots,l_n$  perpendicular to the plane of the  $n$ -gon at  $A_1,A_2,\ldots,A_n$  respectively. Points  $B_1,B_2,\ldots,B_n$  are selected on lines  $l_1,l_2,\ldots,l_n$  respectively so that:

(i)  $B_1,B_2,\ldots,B_n$  are all on the same side of the plane of the  $n$ -gon;
(ii) Points  $B_1,B_2,\ldots,B_n$  lie on a single plane;
(iii)  $A_1B_1=h_1,A_2B_2=h_2,\ldots,A_nB_n=h_n$ .

Express the volume of polyhedron  $A_1A_2\ldots A_nB_1B_2\ldots B_n$  as a function in  $S,h_1,\ldots,h_n$ .

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