
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) The imaginary part of the complex number $z= \frac{(i-1)^{2}+4}{i+1}$ is \_\_\_.

(2) The coefficient of $x^{2}$ in the expansion of $(2x- \frac{1}{2})^{6}$ is \_\_\_\_.

(3) The number of 5-digit numbers formed without repeating digits using the digits $12$, $3$, $4$, $5$, where the number of odd digits is \_\_\_\_ (answer with a number).

(4) Given the hyperbola $\frac{x^{2}}{a^{2}}-y^{2}=1(a > 0)$, if one of its asymptotes has an inclination angle of $30^{0}$, then the eccentricity of this hyperbola is \_\_\_\_.

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