
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 \( f(x) \) defined on \([0,1]\), with \( f(0)=0 \), \( f(1)=1 \), and satisfying the conditions:
(a) For any \( x \in [0,1] \), \( f(x) \geq 0 \);
(b) For any two numbers \( x_1 \geq 0 \), \( x_2 \geq 0 \), and \( x_1 + x_2 \leq 1 \), \( f\left(x_1 + x_2\right) \geq f\left(x_1\right) + f\left(x_2\right) \),

find the smallest positive constant \( c \) such that for any function \( f(x) \) satisfying the above conditions and for all \( x \in [0,1] \), \( f(x) \leq c x \).

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