
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 triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2ac\sin A + a^{2} + c^{2} - b^{2} = 0$.  
$(1)$ If $A = \frac{\pi}{6}$ and $a = 2$, find the area of $\triangle ABC$.  
$(2)$ Find the minimum value of $\frac{4\sin^{2}C + 3\sin^{2}A + 2}{\sin^{2}B}$ and determine the value of $B$ at that point.

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