
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 universal set be $U = \mathbb{R}$, set $A = \{x | 1 \leqslant x \leqslant 5\}$, set $B = \{x | -1-2a \leqslant x \leqslant a-2\}$.  
$(1)$ If $A \cap B = A$, find the range of real number $a$;  
$(2)$ If the proposition "For all $x \in B$, then $x \in A$" is a true proposition, find the range of real number $a$.

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