
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 an arithmetic sequence $\{a\_n\}$ with a non-zero common difference $d$, if $a\_2=5$ and $a\_1$, $a\_3$, $a\_6$ form a geometric sequence.
(1) Find the general term formula of the sequence $\{a\_n\}$;
(2) If sequence $\{b\_n\}$ satisfies $b\_1=0$ and for any $n\geqslant 2$, $|b\_n-b\_{n-1}|=2^{a\_n}$,
$\quad\quad$ (i) Write out all possible values of $b\_3$;
$\quad\quad$ (ii) If $b\_k=2116$, find the minimum value of $k$.

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