
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 $\{a_n\}$ is an arithmetic sequence with a common difference of $d$, and its sum of the first $n$ terms is $S_n$, where $S_4=2S_2+4$. In the sequence $\{b_n\}$, $b_n= \frac{1+a_n}{a_n}$.

$(1)$ Find the value of the common difference $d$;

$(2)$ If $a_1= -\frac{5}{2}$, find the maximum and minimum values in the sequence $\{b_n\}$.

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