
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
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, where $S_n = kn^2 + n$, $n \in \mathbb{N}^*$, and $k$ is a constant.  
(I) Find $a_1$ and $a_n$;  
(II) If for any $m \in \mathbb{N}^*$, $a_m$, $a_{2m}$, and $a_{4m}$ form a geometric sequence, find the value of $k$.

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