
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
Complex numbers \( z_{1}, z_{2}, z_{3}, z_{4}, z_{5} \) satisfy:
\[ 
\begin{cases} 
|z_{1}| \leq 1, |z_{2}| \leq 1 \\ 
|2z_{3} - (z_{1} + z_{2})| \leq |z_{1} - z_{2}| \\ 
|2z_{4} - (z_{1} + z_{2})| \leq |z_{1} - z_{2}| \\ 
|2z_{5} - (z_{3} + z_{4})| \leq |z_{3} - z_{4}| 
\end{cases} 
\]

Find the maximum value of \( |z_{5}| \).

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