
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
Let  $\vartriangle ABC$  be a triangle such that the area $ [ABC] = 10$  and  $\tan (\angle ABC) = 5$ . If the smallest possible value of  $(\overline{AC})^2$  can be expressed as  $-a + b\sqrt{c}$  for positive integers  $a, b, c$ , what is  $a + b + c$ ?


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