
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 set $A=\{x|f(x)=\lg(x-1)+\sqrt{2-x}\}$, and the set $B=\{y|y=2^x+a, x\leq0\}$.
1. If $a=\frac{3}{2}$, find $A\cup B$;
2. If $A\cap B=\emptyset$, find the range of the real number $a$.

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