
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 $P$: real number $x$ satisfies the set $A=\{x\left|\right.-a \lt x \lt 2a\}$, $q$: real number $x$ satisfies the set $B=\{x|x^{2}-5x+4 \lt 0\}$.  
$(1)$ If $a=1$, and both $p$ and $q$ are true propositions, find the range of $x$;  
$(2)$ If $B\subseteq A$, find the range of real number $a$.

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