
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 an acute-angled triangle $\triangle ABC$, with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively, and given that $b^2 \sin C = 4\sqrt{2} \sin B$ and the area of $\triangle ABC$ is $\frac{8}{3}$, find the minimum value of $a^2$.

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