
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  $ABC$  be a right triangle with  $\angle{ACB}=90^{\circ}$ .  $D$  is a point on  $AB$  such that  $CD\perp AB$ . If the area of triangle  $ABC$  is  $84$ , what is the smallest possible value of   $$ AC^2+\left(3\cdot CD\right)^2+BC^2? $$  *2016 CCA Math Bonanza Lightning #2.3*

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