
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 $a$, $b$, and $c$ be positive real numbers with $a \geq b \geq c$. Consider the maximum possible value of $\frac{a}{c}$ for which the system of equations
\[ a^2 + z^2 = c^2 + y^2 = (a - y)^2 + (c - z)^2 \]
has a solution in $(y, z)$ satisfying $0 \leq y < a$ and $0 \leq z < c$. Find $\left(\frac{a}{c}\right)^2$.

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