
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
Given that the trigonometric form of a complex number $z$ is $z=\cos \theta +i\sin \theta $.<br/>$(1)$ If the vector corresponding to the complex number $z$ is $\overrightarrow{OZ}$, and rotating $\overrightarrow{OZ}$ counterclockwise by $15^{\circ}$ results in a vector $\overrightarrow{O{Z}_{1}}$ lying exactly on the positive $y$-axis, find the complex number $z$ (expressed in algebraic form).<br/>$(2)$ If the real part of $z$ is $\frac{r{a}^{2}-1}{1+{a}^{2}}$, does there exist a positive integer $r$ such that $u=|z^{2}+z+1|$ has only a minimum value and no maximum value for any real number $a$? If such a value of $r$ exists, find the value of $a$ that minimizes $u$; if no such value of $r$ exists, explain the reason.

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