
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 \( f(x) = \left\{ \begin{array}{ll} x + \frac{1}{2}, & x \in \left[0, \frac{1}{2}\right) \\ 2(1-x), & x \in \left[\frac{1}{2}, 1\right] \end{array} \right. \), where \( f_{1}(x) = f(x) \) and \( f_{n}(x) = f(f_{n-1}(x)) \), find \( f_{27} \left( \frac{1}{5} \right) \).

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