
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 $g(x)=x^{2}-2ax$ and $f(x)= \frac {1}{3}x^{3}-\ln (x+1)$, if there exists $x\_1\in[0,1]$ and $x\_2\in[1,2]$ such that $f′(x\_1)\geqslant g(x\_2)$ holds, then the range of the real number $a$ is \_\_\_\_\_\_.

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