
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 quadrilateral $ABCD$, $BC=CD=2$ units, $\angle BCD$ is a right angle, $\angle A = \angle D = 120^\circ$, and $AB = 1$ unit. Find the length of diagonal $AC$, which can be expressed as $\sqrt{a} + \sqrt{b}$ units. What is $a + b$?

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