
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 functions $f : [0, 1] \rightarrow \mathbb{R}$ which satisfy


     (i)$f(x)\ge0$ for all $x$ in $[0, 1]$,

     (ii)$f(1) = 1$,

     (iii)     $f(x) + f(y) \le f(x + y)$ whenever $x$, $y$, and $x + y$ are all in $[0, 1]$.
Find, with proof, the smallest constant $c$ such that

$f(x) \le cx$
for every function $f$ satisfying (i)-(iii) and every $x$ in $[0, 1]$.

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