
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 that the first term of the sequence $\{a_{n}\}$ is $1$, the sum of the first $n$ terms of the sequence is $S_{n}$. For a positive geometric sequence $\{b_{n}\}$, it is known that $b_{1}+b_{3}=5$ and $b_{5}-b_{1}=15$. If $S_{n+1}+b_{n}=S_{n}+a_{n}+3$, and in the sequence $\{a_{n}\}$, only $5$ terms are not less than the real number $\lambda$, the range of values for $\lambda$ is ____.

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