
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
In the parallelogram \( OADB \), \( \overrightarrow{OA} = \vec{a} \) and \( \overrightarrow{OB} = \vec{b} \). Points \( M \) and \( N \) are on segments \( AB \) and \( CD \) respectively such that \( BM = \frac{1}{3} BC \) and \( CN = \frac{1}{3} CD \), where \( C \) is the intersection point of diagonals \( AB \) and \( OD \). Express \( \overrightarrow{MN} \) in terms of \( \vec{a} \) and \( \vec{b} \).

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