
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
In a circle with a diameter $\mathrm{AB}=2R$, a symmetric trapezoid $AMNB$ is inscribed.

1. Analyze how the function \(y=\overline{AM}^{2}+\overline{MN}^{2}+\overline{NB}^{2}\) changes as a function of \(M N=x\). Graph this function's variation.

2. Determine \(x\) such that \(y=a^{2}\). Using the previously derived function graph, identify for which values of \(a^{2}\) there are solutions.

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