
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 of four numbers $a_1$, $a_2$, $a_3$, $a_4$, where the first three numbers form a geometric sequence with their sum equal to $k$, and the last three numbers form an arithmetic sequence with their sum equal to $15$, and the common difference is non-zero. For any fixed real number $k$, if the number of sequences that satisfy the conditions is more than $1$, then the range of values for $k$ is ______.

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