
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 sum of the first $n$ terms as $S_n$, and a geometric sequence $\{b_n\}$ whose first term is 1 and common ratio is $q$, it holds that $S_2=a_3=b_3$, 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 $\frac{1}{c_1}, \frac{1}{c_2}, \frac{1}{c_t}$ (where $t\geq3$) form an arithmetic sequence, find the values of $k$ and $t$.

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