
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 $a_1 < 0$ and $a_{n+1}= \frac {a_n}{3a_n+1}$ $(n\in\mathbb{N}^*)$, and another sequence $\{b_n\}$ satisfies $b_n=na_n$ $(n\in\mathbb{N}^*)$. Let $S_n$ be the sum of the first $n$ terms of sequence $\{b_n\}$. If $S_n$ has its minimum value when $n=7$, then the range of values for $a_1$ is \_\_\_\_\_\_.

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