
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
In a sequence $\{a_n\}$ where each term is positive, the sum of the first $n$ terms $S_n=\left( \frac {a_n+1}{2}\right)^2$.

$(1)$ Find the general formula for the sequence $\{a_n\}$;  
$(2)$ If $\frac {1}{a_1a_2}+ \frac {1}{a_2a_3}+\ldots+ \frac {1}{a_na_{n+1}} < k$ always holds, find the range of $k$;  
$(3)$ Is there any positive integer $m$, $k$, such that $a_m$, $a_{m+5}$, $a_k$ form a geometric sequence? If yes, find the values of $m$ and $k$. If no, explain why.

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