
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
Consider the following four propositions:
1. If a function $f(x)$ is increasing for $x > 0$ and also increasing for $x < 0$, then $f(x)$ is an increasing function.
2. If the function $f(x) = ax^2 +bx + 2$ has no intersection with the x-axis, then $b^2 - 8a < 0$ and $a > 0$.
3. The increasing interval for $y = x^2 - 2|x| - 3$ is $[1, +\infty)$.
4. The functions $y = 1 + x$ and $y = \sqrt{(1 + x)^2}$ are equivalent.

The number of correct propositions is_______.

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