
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 six digits $0$, $1$, $2$, $3$, $4$, $5$, complete the following three sub-problems:
1. If digits are allowed to repeat, how many different five-digit even numbers can be formed?
2. If digits are not allowed to repeat, how many different five-digit numbers that are divisible by $5$ and do not have $3$ in the hundreds place can be formed?
3. If $a$ and $b$ in the linear equation $ax + by = 0$ can be any two different digits from the given six digits, how many different lines does the linear equation represent?

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