
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 $A$, in square units, be the area of a trapezoid where the shorter base, the altitude, and the longer base are elements of a geometric progression with the shorter base being the first term. If the shorter base has the length $a$ and the common ratio of the progression is $r$, find the area of the trapezoid.
**A)** $\frac{1}{2} a^2 r^2$
**B)** $\frac{1}{2} a^2 r (1 + r)$
**C)** $\frac{1}{2} a^2 r (1 + r^2)$
**D)** $a^2 r (1 + r^2)$
**E)** $ar^2$

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