
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 functions $f(x)=x^{2}+2$ and $g(x)=|x-a|-|x-1|$, where $a\in \mathbb{R}$,  
(1) If $a=4$, find the solution set of the inequality $f(x) > g(x)$;  
(2) If for any $x_{1}, x_{2} \in \mathbb{R}$, the inequality $f(x_{1}) \geq g(x_{2})$ always holds, find the range of the real number $a$.

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