
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 \( z_{i} \in \mathbf{C} \) (i = 1,2, \ldots, 5), \(\left|z_{1}\right| \leq 1\), \(\left|z_{2}\right| \leq 1\) and \(\mid 2 z_{3}-\left(z_{1} + z_{2}\right) \mid \leq \left| z_{1} - z_{2} \right|\), \(\mid 2 z_{4} - \left(z_{1} + z_{2}\right) \mid \leq \left| z_{1} - z_{2} \right|\),\(\mid 2 z_{5} - \left(z_{3} + z_{4}\right) \mid \leq \left| z_{3} - z_{4} \right| \). Find the maximum value of \(\left|z_{5}\right|\).

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