
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 \(\mathbf{R} \) (the set of real numbers), it satisfies the following conditions:
1. \( f(1) = 1 \)
2. When \( 0 < x < 1 \), \( f(x) > 0 \)
3. For any real numbers \( x \) and \( y \), the equation \( f(x+y) - f(x-y) = 2 f(1-x) f(y) \) holds

Find \( f\left(\frac{1}{3}\right) \).

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