
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 common difference $d$, and $\{b_n\}$ is a geometric sequence with common ratio $q$, where $q \neq \pm1$, and the set of positive integers $E=(m,p,r)$ with $m < p < r$:

(1) If $a_1+b_2=a_2+b_3=a_3+b_1$, find the value of $q$;  
(2) If the three numbers in set $E$ form an arithmetic sequence with a common difference greater than $1$, and $a_m+b_p=a_p+b_r=a_r+b_m$, find the maximum value of $q$;  
(3) If $b_n=(-\frac{1}{2})^{n-1}$, and $a_m+b_m=a_p+b_p=a_r+b_r=0$, try to write out a set $E$ that satisfies the conditions and the corresponding general formula for $a_n$. (Note: It is not necessary to write out the solution process for this part.)

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