
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 $f(x)=|x+1-2a|+|x-a^{2}|$, $g(x)=x^{2}-2x-4+\dfrac{4}{(x-1)^{2}}$.

$(1)$ Find the solution set of the inequality $f(2a^{2}-1) > 4|a-1|$;

$(2)$ If there exist real numbers $x$, $y$ such that $f(x)+g(y)\leqslant 0$ holds, find the range of values for the real number $a$.

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