
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 trapezoid with side lengths 4, 6, 8, and 10, calculate the sum of all possible areas of this trapezoid which can be formed with these sides, and express the answer in the form of $r_1\sqrt{n_1}+r_2$, where $r_1$ and $r_2$ are rational numbers and $n_1$ is a positive integer not divisible by the square of any prime.

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