
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

Let \(A\) be the vertex of a given angle \(\varphi\). On one side of this angle \(AX\), we select points \(A_{1}, A_{2}, A_{3}, \ldots A_{n}\) such that \(A A_{1}=a\), \(A_{1} A_{2}=2a\), \(A_{2} A_{3}=3a\), \ldots, \(A_{n-1} A_{n}=na\). Then, at points \(A_{1}, A_{2}, A_{3}, \ldots A_{n}\), we draw perpendiculars to \(AX\) which intersect the other side of the angle \(\varphi\) at points \(B_{1}, B_{2}, B_{3}, \ldots B_{n}\). What is the area of:

(1) the trapezoid \(A_{n-1} A_{n} B_{n} B_{n-1}\) and

(2) the triangle \(A A_{n} B_{n}\)?

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