
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
Here are four propositions about necessary and sufficient conditions:
1. The necessary and sufficient condition for vector $\mathbf{b}$ to be collinear with a nonzero vector $\mathbf{a}$ is that there exists exactly one real number $\lambda$ such that $\mathbf{b} = \lambda\mathbf{a}$;
2. The necessary and sufficient condition for $a$, $b$, $c$ to form a geometric sequence is $b^2 = ac$;
3. Two events being mutually exclusive is the necessary and sufficient condition for these two events to be complementary;
4. The necessary and sufficient condition for the function $y = x^2 + bx + c$ to be an even function is $b = 0$.

Among these, the numbers of the true propositions are: (Write down the numbers of all true propositions)

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