
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 $$f(x)= \begin{cases} \overset{x+2,x<a}{x^{2},x\geq a}\end{cases}$$, find the range of values for the real number $a$ such that the function $f(x)$ is monotonic on $\mathbb{R}$. Also, find the range of values for $a$ such that for any real number $x_1 < a$, there exists a real number $x_2 \geq a$ that satisfies $f(x_1) + f(x_2) = 0$.

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