
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
Find the values of the real numbers $a$ and $b$ in the solution set $(-1, b)$ for the inequality about $x$: $$\left|\begin{vmatrix} x+a & 2 \\ 1 & x \end{vmatrix}\right| < 0$$

Also, if $z_1 = a + bi$ and $z_2 = \cos{\alpha} + i\sin{\alpha}$, and $z_1z_2$ is a purely imaginary number, find the value of $\tan{\alpha}$.

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