
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
Selected Exercise $(4-5)$: Inequality Lecture

Given that the function $f(x) = m - |5 - 2x| - |2x - 1|$ has a zero point on $R$.

$(1)$ Find the range of values for the real number $m$;

$(2)$ When $m$ takes the minimum value, solve the inequality regarding $x$: $|x - 3| + |x + m| \leqslant 2m$.

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