
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 that the solution set for the inequality $ax^2+ax+2>0$ is $\mathbb{R}$ (the set of all real numbers), let the set of all numerical values of the real number $a$ be denoted as $M$.
(1) Find the set $M$.
(2) If $t>0$, for all $a \in M$, it holds that $(a^2-2a)t \leq t^2 + 3t - 46$. Find the minimum value of $t$.

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