
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
If the plane vectors \(\vec{a}, \vec{b}, \vec{c}\) satisfy \(\vec{a} \cdot (\vec{a} + \vec{c}) = 0\) and \(|\vec{a} + \vec{b} - 2\vec{c}| = 2\), find the maximum value of \(\vec{a} \cdot \vec{b}\).

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