
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
Let $\{a_n\}$ be an arithmetic sequence with the first term $1$ and common difference $2$, and $\{b_n\}$ be a geometric sequence with the first term $1$ and common ratio $q$. Define $c_n=a_n+b_n$ for $n=1, 2, 3, \ldots$.

$(1)$ If $\{c_n\}$ is an arithmetic sequence, find the value of $q$;  
$(2)$ Find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $S_n$.

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