
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 that $x$ is a real number, and the complex number $z=(x^2+x-2)+(x^2+3x+2)i$ is purely imaginary, then $x=\boxed{1}$.

(2) Using the digits $4$ and $5$ to form a four-digit number, with both $4$ and $5$ appearing at least once, there are $\boxed{14}$ such four-digit numbers.

(3) If $\int_{0}^{x}{a^2da=x^2}$ $(x > 0)$, then $\int_{1}^{x}{|a-2|da}=\boxed{1}$.

(4) If the function $f(x)=x^3-3x+5-a$ $(a\in \mathbb{R})$ has $2$ zeros in the interval $\left(-3,\frac{3}{2}\right)$, then the range of $a$ is $\boxed{\{3\}\cup\left(\frac{31}{8},7\right)}$.

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