
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 a function $f\left(x\right)=a^{x}+b\left(a \gt 0,a\neq 1\right)$.  
$(1)$ If the graph of $f\left(x\right)$ passes through the point $\left(0,2\right)$, find the value of $b$.  
$(2)$ If the maximum value of the function $f\left(x\right)$ on the interval $\left[2,3\right]$ is greater than the minimum value by $\frac{{a^2}}{2}$, find the value of $a$.

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