
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
Elective 4-5: Special Topics on Inequalities

Given the function $f(x)=|x-a|$.

$(1)$ If the solution set of the inequality $f(x) \leqslant 2$ is $\{x|1 \leqslant x \leqslant 5\}$, find the value of the real number $a$;

$(2)$ Under the condition of $(1)$, if the inequality $f(2x)+f(x+2) \geqslant m$ holds true for all real numbers $x$. Find the range of values for the real number $m$.

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