
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 $f(x)=-x^{2}+2\ln x$ has its maximum value equal to $-\frac{1}{2}$ times the minimum value of the function $g(x)=x+ \frac {a}{x}$, and for all $x_{1},x_{2}\in\left[ \frac {1}{e},3\right]$, the inequality $\frac {f(x_{1})-g(x_{2})}{k-1}\leqslant 1$ always holds, determine the range of the real number $k$.

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