
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 the set $A = [\tan(-\frac{19\pi}{6}), \sin(-\frac{19\pi}{6})]$; If the function $f(x) = \sqrt{x^2 + mx + m}$ has the domain $\mathbb{R}$, let the set of possible values for the real number $m$ be $B$, and the set $C = \{x | a + 1 < x < 2a\}$, where $a$ is a real number.
(1) Find the sets $A$, $B$, and the union of $A$ and $B$ ($A \cup B$).
(2) If $C \subseteq (A \cup B)$, determine the range of values for $a$.

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