
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 condition p: $|x-1| > a$ ($a \geq 0$) and condition q: $\lg(x^2-3x+3) > 0$,  
(1) Find the solution set of the inequality that satisfies conditions p and q.  
(2) Using the given two conditions as A and B, construct the proposition: "If A, then B". Is there a non-negative real number $a$ that makes the original proposition true and its converse false? If it exists, find the range of $a$. If not, explain why.

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