
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 following four propositions:
- (1) $0$ is greater than $-i$;
- (2) Two complex numbers are conjugates of each other if and only if their sum is a real number;
- (3) The necessary and sufficient condition for $x+yi=1+i$ is $x=y=1$;
- (4) If we associate the real number $a$ with $ai$, then there is a one-to-one correspondence between the set of real numbers and the set of pure imaginary numbers.

The number of true propositions is ______.

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