
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
In convex quadrilateral $ABCD$, $AB=10$, $BC=6$, $CD=DA=12$, and $\angle CDA=90^\circ$. Determine the area of $ABCD$ if it can be written in the form $\sqrt{a}+b\sqrt{c}$, where $a$ and $c$ have no perfect square factors greater than 1. What is $a+b+c$?

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