
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 sequence $\{a_n\}$ is an arithmetic sequence with all positive terms, and $S_n$ is the sum of its first $n$ terms, and $4S_n=(a_n+1)^2$.

$(1)$ Find the values of $a_1$, $a_2$, and the general formula for $\{a_n\}$;

$(2)$ Find the minimum value of $S_n-\frac{7}{2}a_n$.

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