
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 convex quadrilateral $ABCD$ with $AB = 20$, $\angle A = 90^\circ$, and $\overline{AB} \parallel \overline{CD}$, in some order, the lengths of the four sides form an arithmetic progression, and the side $\overline{AB}$ is the side of maximum length. If $a$ is the length of another side, calculate the sum of all possible values of $a$.

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