
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 common difference equal to the common ratio of a geometric sequence $\{b_{n}\}$, where $a_{1}=1$, $b_{2}=4$, and $b_{3}-a_{6}=-3$. Find:<br/>
$(1)$ The general formulas for $\{a_{n}\}$ and $\{b_{n}\}$;<br/>
$(2)$ Let $\{c_{n}\}$ be a new sequence formed by arranging the terms of $\{a_{n}\}$ and $\{b_{n}\}$ in ascending order. Calculate the sum $S_{60}$ of the first 60 terms of the sequence $\{c_{n}\}$.

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