
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
For a sequence of integers $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and it satisfies $S_n= \frac {1}{4}a_n^{2}+ \frac {1}{2}a_n+ \frac {1}{4}$ for $n\in\mathbb{N}^{+}$.  
$(1)$ Find $a_n$;  
$(2)$ Suppose the sequence $\{a_n+b_n\}$ is a geometric sequence with the first term being $1$ and the common ratio being $q$, find the sum of the first $n$ terms $S_n$ of the sequence $\{b_n\}$.

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