
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 non-degenerate triangle with perimeter  $4$  such that  $a=bc\sin^2A$ . If  $M$  is the maximum possible area of  $ABC$  and  $m$  is the minimum possible area of  $ABC$ , then  $M^2+m^2$  can be expressed in the form  $\frac{a}{b}$  for relatively prime positive integers  $a$  and  $b$ . Compute  $a+b$ .

*2016 CCA Math Bonanza Lightning #4.3*

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