
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
Let \( f \) be a function on \([0,1]\) satisfying:
1. \( f(0) = 0, f(1) = 1 \).
2. For all \( x, y \in [0, 1] \), with \( x \leq y \), \( f\left(\frac{x+y}{2}\right) = (1-a) f(x) + a f(y) \), where \( a \) is a real number and \( 0 \leq a \leq 1 \). Find \( f\left(\frac{1}{7}\right) \).

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