
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
Proposition $p$: The equation $x^{2}-x+a^{2}-6a=0$ has one positive root and one negative root.
Proposition $q$: The graph of the function $y=x^{2}+(a-3)x+1$ intersects with the $x$-axis.
Determine the range of real numbers $a$ such that the proposition "$p \vee q$" is true, while "$p \wedge q$" is false.

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