
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 $f(x)=|x-1|+|x+a^{2}|$, where $a\in\mathbb{R}$.  
(1) When $a= \sqrt {2}$, find the solution set of the inequality $f(x)\geqslant 6$;  
(2) If there exists $x_{0}\in\mathbb{R}$ such that $f(x_{0}) < 4a$, find the range of the real number $a$.

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