
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  $ a \in \mathbb{R}, 0 < a < 1,$  and  $ f$  a continuous function on  $ [0, 1]$  satisfying  $ f(0) \equal{} 0, f(1) \equal{} 1,$  and

\[ f \left( \frac{x\plus{}y}{2} \right) \equal{} (1\minus{}a) f(x) \plus{} a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.\]

Determine  $ f \left( \frac{1}{7} \right).$ 

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