
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 the function $y=b\cdot a^{x}$ ($a$ and $b$ are constants, $a \gt 0$, and $a \neq 1$).  
$(1)$ If the graph of the function passes through points $A(1,6)$ and $B(3,24)$, find the values of the real numbers $a$ and $b$.  
$(2)$ If the function is an exponential function, and the difference between the maximum and minimum values on the interval $[-1,1]$ is $1$, find the expression of the function.

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