
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

Let the function $f(x) \leqslant |2x-1|, x \in \mathbb{R}$.

$(1)$ Solve the inequality $f(x) \leqslant 5 - f(x-1)$.

$(2)$ Given that the solution set of the inequality $f(x) \leqslant f(x+a) - |x-a|$ is $M$, and if $(\frac{1}{2},1) \subseteq M$, find the range of the real number $a$.

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