
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
Given $f(x)=|x+a|$ and $g(x)=|x+3|-x$, let the solution set of the inequality $f(x) < g(x)$ with respect to $x$ be denoted as $M$.  
$(1)$ If $a-3 \in M$, find the range of the real number $a$;  
$(2)$ If $[-1,1] \subseteq M$, find the range of the real number $a$.

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