
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) Let $a > 0$. If the area enclosed by the curve $y=\sqrt{x}$ and the lines $x=a$, $y=0$ is $a^{2}$, then $a=$ _______.

(2) Given $a=\dfrac{-3-i}{1+2i}$, then $a^{4}=$ _______.

(3) Let $O$ be the origin, and the complex numbers corresponding to vectors $\overrightarrow{OA}$, $\overrightarrow{OB}$ are $2-3i$ and $-3+2i$, respectively. Then, the complex number corresponding to vector $\overrightarrow{BA}$ is _______.

(4) The slope of the tangent line to the curve $y={{e}^{\frac{1}{2}x}}$ at the point $(4,e^{2})$ is _______.

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