
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
For the functions $f(x)$ and $g(x)$, denote the set $D_{f>g} = \{x | f(x) > g(x)\}$.
1. Given $f(x) = 2|x|$, $g(x) = x + 3$, find $D_{f>g}$.
2. Let $f_1(x) = x - 1$, $f_2(x) = (\frac{1}{3})^x + a \cdot 3^x + 1$, $h(x) = 0$. If $D_{f_1>h} \cup D_{f_2>h} = \mathbb{R}$, find the range of the real number $a$.

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