
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 function $f(x) = 1 + ax$, where $f(x) \geq 0$ holds true for any $x \in (0, 2]$, and function $g(x) = ax^2 + 2x + a$ has zero points on $(0, +\infty)$.

1. Find the range of real number $a$ if the proposition $p$ is true.
2. Find the range of real number $a$ if "$p$ or $q$" is true and "$p$ and $q$" is false.

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