
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 proposition $p: \exists x \in \mathbb{R}$, $ax^{2} + ax + 1 = 0$, proposition $q: Given m > 0, \forall x \in (-1, +\infty), x - a + \frac{m}{x+1} \geq 0$.  
$(1)$ If proposition $p$ is true, find the range of values for $a$;  
$(2)$ If "proposition $p$ is true" is a necessary but not sufficient condition for "proposition $q$ is true", find the range of values for $m$.

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