
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
Find all real numbers  $c$  such that there exists a function  $f: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$  which satisfies the following.

For all nonnegative reals  $x, y$ ,  $f(x+y^2) \ge cf(x)+y$ .

Here  $\mathbb{R}_{\ge 0}$  is the set of all nonnegative reals.

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