
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 $F$: "$\frac{2a+1}{a} \leqslant 1$"; $F$: "There exists $x \in [0,2)$, such that $-x^3 + 3x + 2a - 1 = 0$";

(Ⅰ) Find the range of real numbers $a$ for which $F$ and $F$ are true propositions respectively;

(Ⅱ) If "F or F" is a true proposition, find the range of real numbers $a$.

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