
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
The following statements are given:
1) Joe is skillful;
2) Joe is not lucky;
3) Joe is lucky, but he is not skillful;
4) If Joe is skillful, then he is not lucky;
5) Joe is skillful if and only if he is lucky;
6) Joe is either skillful or lucky, but not both at the same time.

What is the greatest number of these six statements that can be true simultaneously?

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