
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 $f(x)=\log_{2}(|x+1|+|x-1|-a)$,  
(I) when $a=3$, determine the domain of the function $f(x)$;  
(II) If the solution set of the inequality $f(x) \geqslant 2$ is $\mathbb{R}$ (the set of all real numbers), find the maximum value of the real number $a$.

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