
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 function $f(x) = x^2 + ax + 3$,
(1) when $x \in \mathbb{R}$, $f(x) \geq 2$ always holds, find the range of values for $a$;
(2) when $x \in \mathbb{R}$, let $g(x) = f(2^x)$.
   ① Find the range of $g(x)$;
   ② If there exist solutions for $g(x) \leq a$, find the range of values for $a$.

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