
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 function $f(x) = x^2 - |x+a|$, among the following propositions:<br/>① For all $a \in \mathbb{R}$, $f(x)$ is not a monotonic function on $\mathbb{R}$;<br/>② There exists $a \in \mathbb{R}$ such that $f(x)$ is an even function on $\mathbb{R}$;<br/>③ If the minimum value of $f(x)$ is $-\frac{5}{4}$, then $a=-1$;<br/>④ There exists $a \lt 0$ such that $f(x)$ has three zeros.<br/>The correct propositions are ____.

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