
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 points $A(0,1)$ and $B(3,2)$, and vector $\overrightarrow{AC}=(-4,-3)$, determine the value of vector $\overrightarrow{BC}$.
2. A line $l: \frac{x}{a} + \frac{y}{b} = 1 (a > 0, b > 0)$ passes through point $(3,2)$. Determine the minimum value of $a + b$.
3. In a closed right triangular prism $ABC-A_{1}B_{1}C_{1}$, there is a sphere with volume $V$. Given that $AB \perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$, determine the maximum value of $V$.
4. A function $f(x)$ is defined as:
$$
f(x) =
\begin{cases}
x\ln (1+x)+{x}^{2}, & x \geq 0 \\
-x\ln (1-x)+{x}^{2}, & x < 0
\end{cases}
$$
Determine the range of real numbers $a$ such that $f(-a) + f(a) \leq 2f(1)$.

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