
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
The function $f(x)= \begin{cases} a(x-1)^{2}+1,x < 1 \\ (a+3)x+4a,x\geqslant 1 \end{cases}$ satisfies $\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}} > 0$ for any $x_{1} < x_{2}$. Determine the range of values for $a$.

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