
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 $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ be vectors, with $\mathbf{m}$ being the midpoint of $\mathbf{a}$ and $\mathbf{b}$. Given $\mathbf{m} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}$, $\mathbf{a} \cdot \mathbf{b} = 12$, and $\mathbf{c} \cdot (\mathbf{a} + \mathbf{b}) = 0$, find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2$.

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