
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 set $A = \{x | (x-4)(x-a) = 0, a \in \mathbb{R}\}$, and set $B = \{x | (x-1)(x-4) = 0\}$.

(1) Find $A \cup B$, $A \cap B$;  
(2) If $A \subseteq B$, find the value of the real number $a$;  
(3) If $a = 5$, then the total number of proper subsets of $A \cup B$ is \_\_\_\_\_\_, and for the set $P$ satisfying $(A \cap B) \not\subseteq P \not\subseteq (A \cup B)$, list all possible sets $P$.

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