
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
Let the solution set of the inequality $4^{x}-5\cdot 2^{x}+4 \lt 0$ be $A$, $B=\{x|3-2m \lt x \lt m+1\}$.  
$(1)$ Find the set $A$;  
$(2)$ If $B\subseteq A$, find the range of real numbers $m$.

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