
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 a quadratic inequality in $x$, $ax^{2}-(a+1)x+b < 0$, its solution set is $\{x|x < - \frac{1}{2} \text{ or } x > 1\}$.

(Ⅰ) Find the values of $a$ and $b$;

(Ⅱ) If the inequality $bx^{2}+(m+2a)x+3-m\geqslant 0$ holds for any real number $m \in [0,4]$, find the range of the real number $x$.

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