
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 $f(x)= \begin{vmatrix} ax & x \\ -2 & 2x \end{vmatrix}$ ($a$ is a constant), and $g(x)= \frac {2x^{2}+1}{x}$, it is known that for $x_{1}$, $x_{2}∈[1,4]$, we always have $f(x_{1})≤g(x_{2})$. Determine the range of values for the real number $a$.

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