
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
The solution set for the inequality concerning $x$, $| \begin{vmatrix} x+a & 2 \\ 1 & x \end{vmatrix} | < 0$, is $(-1, b)$.
(1) Find the values of the real numbers $a$ and $b$;
(2) Given $z_1 = a + bi$ and $z_2 = \cos\alpha + i\sin\alpha$, and that $z_1z_2$ is a pure imaginary number, find the value of $\tan\alpha$.

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