
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 complex numbers $\mathrm{z}_{\mathrm{i}} (\mathrm{i} = 1, 2, 3, 4, 5)$ satisfying: 
$$\left\{\begin{array}{c}
\left|z_{1}\right| \leq 1, \quad \left|z_{2}\right| \leq 1 \\
\left|2 z_{3}-(\mathrm{z}_{1}+\mathrm{z}_{2})\right| \leq \left|\mathrm{z}_{1}-\mathrm{z}_{2}\right| \\
\left|2 \mathrm{z}_{4}-(\mathrm{z}_{1}+\mathrm{z}_{2})\right| \leq \left|\mathrm{z}_{1}-\mathrm{z}_{2}\right| \\
\left|2 z_{5}-(\mathrm{z}_{3}+\mathrm{z}_{4})\right| \leq \left|\mathrm{z}_{3}-\mathrm{z}_{4}\right|
\end{array} \right.$$
Find the maximum value of $\left|\mathrm{z}_{5}\right|$.

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