
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 the sum of the first $n$ terms denoted as $S_n$, and a geometric sequence $\{b_n\}$ with the first term $1$ and common ratio $q$, where $S_2=a_3=b_3$, and $a_1$, $a_3$, $b_4$ form a geometric sequence.  
$(1)$ Find the general formulas for $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Let $c_n=k+a_n+\log_{3}b_n$ ($k\in \mathbb{N}^{+}$), if $\frac{1}{c_1}$, $\frac{1}{c_2}$, $\frac{1}{c_t}$ ($t\geqslant 3$) form an arithmetic sequence, find the values of $k$ and $t$.

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