
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 a function \( f(x) \) defined on \([0,1]\), where \( f(0)=0 \), \( f(1)=1 \), and satisfying the following conditions:
(a) For any \( x \in [0,1] \), \( f(x) \geq 0 \);
(b) For any two real numbers \( x_1, x_2 \) satisfying \( x_1 \geq 0, x_2 \geq 0, x_1 + x_2 \leq 1 \), \( f(x_1 + x_2) \geq f(x_1) + f(x_2) \).

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

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