
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 the propositions \( p \): The inequality \( (a-2)x^2+2(a-2)x-4 < 0 \) holds \( \forall x \in \mathbb{R} \); proposition \( q \): One root of the equation \( x^2+(a-1)x+1=0 \) is in the interval \( (0,1) \) and the other root is in the interval \( (1,2) \). If \( p \vee q \) is a true proposition and \( p \wedge q \) is a false proposition, find the range of the real number \( a \).

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