
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
Determine the smallest positive real number  $r$  such that there exist differentiable functions  $f: \mathbb{R} \rightarrow \mathbb{R}$  and  $g: \mathbb{R} \rightarrow \mathbb{R}$  satisfying 
(a)  $f(0)>0$ , 
(b)  $g(0)=0$ , 
(c)  $\left|f^{\prime}(x)\right| \leq|g(x)|$  for all  $x$ , 
(d)  $\left|g^{\prime}(x)\right| \leq|f(x)|$  for all  $x$ , and 
(e)  $f(r)=0$ .

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