
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 $k$ and $b$ be non-zero constants. Consider the following three conditions:
1. Both sequences $\{a_n\}$ and $\{ka_n+b\}$ are geometric progressions.
2. Sequence $\{a_n\}$ is an arithmetic progression, while $\{ka_n+b\}$ is a geometric progression.
3. Sequence $\{a_n\}$ is a geometric progression, while $\{ka_n+b\}$ is an arithmetic progression.

One of these conditions necessarily implies that the sequence $\{a_n\}$ is a constant sequence. Determine which condition(s) lead to this conclusion by listing their corresponding numbers.

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