
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
In the sequence $\{a_n\}$, if for any $n \in N^*$, it holds that $\dfrac{a_{n+2}}{a_{n+1}} - \dfrac{a_{n+1}}{a_{n}} = \lambda$ (where $\lambda$ is a constant), then the sequence $\{a_n\}$ is called a ratio-difference sequence, and $\lambda$ is called the ratio difference. Given the following propositions:  
$(1)$ If the sequence $\{F_n\}$ satisfies $F_1=1$, $F_2=1$, and $F_n=F_{n-1}+F_{n-2}$ $(n \geqslant 3)$, then this sequence is not a ratio-difference sequence;  
$(2)$ If the sequence $\{a_n\}$ satisfies $a_n=(n-1) \cdot 2^{n-1}$, then the sequence $\{a_n\}$ is a ratio-difference sequence, and the ratio difference $\lambda=2$;  
$(3)$ A geometric sequence is always a ratio-difference sequence, but an arithmetic sequence is not necessarily a ratio-difference sequence;  
$(4)$ If $\{a_n\}$ is an arithmetic sequence and $\{b_n\}$ is a geometric sequence, then the sequence $\{a_nb_n\}$ is a ratio-difference sequence.  
Among these, the true propositions are __________.

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