
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 $f(x) = |x-1| - |x+3|$.

$(1)$ Solve the inequality $f(x) > 2$;

$(2)$ If the inequality $f(x) \leqslant kx + 1$ always holds for $x \in [-3, -1]$, find the range of the real number $k$.

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