
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 differentiable functions  $f, g:[0,\infty) \to \mathbb{R}$  and the real constant  $k\geq 0$  such that 
\begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt  
g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*} 
and  $f(0)=k, f'(0)=-k^2/3$  and also  $f(x)\neq 0$  for all  $x\geq 0$ .  
*(Nora Gavrea)*

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