
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 a sequence $\{a_n\}$ where each term satisfies: $a_1 = 1 - 3k$ ($k \in \mathbb{R}$), $a_n = 4^{n-1} - 3a_{n-1}$

(1) Determine whether the sequence $\left\{a_n - \frac{4^n}{7}\right\}$ is a geometric sequence;  
(2) Find the general formula for the sequence $\{a_n\}$;  
(3) If the sequence $\{a_n\}$ is increasing, find the range of values for $k$.

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