
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 set $A = \{x | -1 \leqslant x \leqslant 2\}$, and $B = \{x | x^2 - (2m + 1)x + 2m < 0\}$.

$(1)$ When $m < \frac{1}{2}$, simplify the set $B$;

$(2)$ If $A \cup B = A$, find the range of the real number $m$.

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