
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 four numbers $a_1$, $a_2$, $a_3$, and $a_4$ form a geometric sequence in order, and the common ratio $q$ is not equal to 1. Removing one number from this sequence to form a new sequence (keeping the original order) results in an arithmetic sequence. Determine the set of possible values for the positive number $q$.

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