
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
Given that \(\overrightarrow{A C}\) is the sum of \(\overrightarrow{B A}\) and \(\overrightarrow{B D}\), and \(\overrightarrow{B D}\) is the sum of \(\overrightarrow{A D}\) and \(\frac{2}{3} \overrightarrow{A B}\). If \(\overrightarrow{A C} = \vec{a}\) and \(\overrightarrow{B D} = \vec{b}\), express \(\overrightarrow{A B} + \overrightarrow{A D}\) in terms of \(\vec{a}\) and \(\vec{b}\).

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