
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 the angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \) be \(\frac{\pi}{3}\). The angle between vectors \(\mathbf{c} - \mathbf{a}\) and \(\mathbf{c} - \mathbf{b}\) is \(\frac{2\pi}{3}\). Given that \(|\mathbf{a} - \mathbf{b}| = 5\) and \(|\mathbf{c} - \mathbf{a}| = 2\sqrt{3}\), find the maximum value of \(\mathbf{a} \cdot \mathbf{c}\).

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