
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 $f(x)=x^{2}+x+2a$ and $g(x)=-\frac{1}{x}$, if there exist two points $A(x_{1},f(x_{1}))$ and $B(x_{2},g(x_{2}))$ ($x_{1} < 0, x_{2} > 0$) such that the line $AB$ is tangent to both the graphs of $y=f(x)$ and $y=g(x)$, determine the range of the real number $a$.

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