
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 the function $f(x)=|x-a|+ \frac{1}{2a}$ (where $a\neq 0$),  
(1) If the inequality $f(x)-f(x+m)\leqslant 1$ always holds, find the maximum value of the real number $m$;  
(2) When $a < \frac{1}{2}$, the function $g(x)=f(x)+|2x-1|$ has a zero, determine the range of the real number $a$.

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