
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 proposition $p$: real number $x$ satisfies $M=\{x | -2 \leqslant x \leqslant 5\}$, proposition $q$: real number $x$ satisfies $N=\{x | 1-2m \leqslant x \leqslant 2+m\}$.  
$(1)$ If the proposition "$\forall x \in M$, $x \in N$" is a true proposition, find the range of real number $m$;  
$(2)$ If proposition $p$ is a necessary but not sufficient condition for proposition $q$, find the range of real number $m$.

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