
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 that the point $(2,9)$ lies on the graph of the function $f(x)=a^{x}$ where $a>0$ and $a\neq 1$, for any $x_{1}$, $x_{2}$ in the domain of the function $y=f(x)$, the following conclusions hold:<br/>①$f(x_{1}+x_{2})=f(x_{1})\cdot f(x_{2})$;<br/>②$f(x_{1}\cdot x_{2})=f(x_{1})+f(x_{2})$;<br/>③$\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}} \lt 0$;<br/>④$f(\frac{x_{1}+x_{2}}{2}) \lt \frac{f(x_{1})+f(x_{2})}{2}$<br/>The correct numbers of the above conclusions are ____.

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