
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
On the complex plane, the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z} + 1$ has an area of $\frac{24}{25}$. If the real part of $z$ is positive, let $d$ be the smallest possible value of $\left| z + \frac{1}{z} + 1 \right|.$ Compute $d^2.$

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