
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
(1) Given the function $f(x) = \log_2(x^2 + a)$, if $f(3) = 1$, find the value of $a$.

(2) If $x$ and $y$ satisfy the constraints $\begin{cases} x - 2y - 2 \leqslant 0 \\ x - y + 1 \geqslant 0 \\ y \leqslant 0 \end{cases}$, find the maximum value of $z = 3x + 2y$.

(3) The line $y = x + 1$ intersects the circle ${x^2} + {y^2} + 2y - 3 = 0$ at points $A$ and $B$. Find the length of $|AB|$.

(4) In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $b\sin C + c\sin B = 4a\sin B\sin C$ and ${b^2} + {c^2} - {a^2} = 8$, find the area of $\triangle ABC$.

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