
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 $A = \{x \mid -x^{2} + 12x - 20 \gt 0\}$, set $B = \{x \mid 5-a \lt x \lt a\}$, where $a \in \mathbb{R}$.  
$(1)$ If "$x \in A$" is a necessary condition for "$x \in B$", find the range of $a$;  
$(2)$ If $A \cap B \neq \varnothing$, find the range of $a$.

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