
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
Let the function be defined as:
$$
f(x) = \min \left\{x^2 - 1, x + 1, -x + 1\right\}
$$
where $\min \{x, y, z\}$ denotes the minimum of $x$, $y$, and $z$. If $f(a+2) > f(a)$, determine the range of the real number $a$.

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