
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 an increasing geometric sequence $\{a_n\}$, the product of its first three terms is 8, and these three terms, when individually increased by 1, 2, and 2 respectively, form an arithmetic sequence.

(1) Find the general term formula for the geometric sequence $\{a_n\}$;
(2) If the inequality $a_n^2 + 2^n a_n - k \geq 0$ holds for all $n \in \mathbb{N}^*$, find the range of values for the real number $k$.

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