
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 $f(x) = x^2 + (a+1)x + \lg|a+2|$ ($a \in \mathbb{R}$, and $a \neq -2$).

(I) If $f(x)$ can be expressed as the sum of an odd function $g(x)$ and an even function $h(x)$, find the expressions for $g(x)$ and $h(x)$.

(II) Let proposition P be: The function $f(x)$ is increasing on the interval $[(a+1)^2, +\infty)$; let proposition Q be: The function $g(x)$ is a decreasing function. If exactly one of propositions P and Q is true, find the range of values for $a$.

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