
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 \( ABC \) be a triangle with \( AB = 5 \), \( BC = 4 \), and \( AC = 3 \). Let \(\mathcal{P}\) and \(\mathcal{Q}\) be squares inside \(ABC\) with disjoint interiors such that they both have one side lying on \(AB\). Also, the two squares each have an edge lying on a common line perpendicular to \(AB\), and \(\mathcal{P}\) has one vertex on \(AC\) and \(\mathcal{Q}\) has one vertex on \(BC\). Determine the minimum value of the sum of the areas of the two squares.

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