
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 a function \( f(x) \) on the interval \([0,1]\) that satisfies the following conditions:
(1) \( f(x) \geq 0 \), for any \( x \in [0,1] \);
(2) \( f(1)=1 \);
(3) \( f(x) + f(y) \leq f(x+y) \), for \( x, y, x+y \in [0,1] \).

Find the smallest constant \( c \) such that for any function \( f(x) \) satisfying conditions (1)-(3), \( f(x) \leq cx \), for all \( x \in [0,1] \).

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