
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 that the even function $f(x)$ defined on $\mathbb{R}$ is decreasing on $[0,+\infty)$, and the inequality $f(-ax+x^{3}+1)+f(ax-x^{3}-1)\geqslant 2f(1)$ holds for $x\in(0, \sqrt {2}]$, determine the range of values for the real number $a$.

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