
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 hexagon $ABCDEF$, $AB = BC = CD = DE = EF = 2$ units, $\angle F$ is a right angle, and $m\angle A = m\angle B = m\angle C = m\angle D = m\angle E = 120^\circ$. The length of segment $AF$ can be expressed in the simplest radical form as $a + b\sqrt{c}$ units. What is the value of $a+b+c$?

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