
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 a regular triangle  $ OAB$  with side length  $ 1$ . Let  $ M,\ N$  be the point which divides equally  $ AB$  into three parts,  $ M$  is the closest to  $ A$ .
Denote  $ \overrightarrow{OA} \equal{} \overrightarrow{a},\ \overrightarrow{OM} \equal{} \overrightarrow{m},\ \overrightarrow{ON} \equal{} \overrightarrow{n},\ \overrightarrow{OB} \equal{} \overrightarrow{b}$ .

(1) Find the real numbers  $ x,\ y$  such that  $ \overrightarrow{m} \equal{} x \overrightarrow{a} \plus{} y \overrightarrow{b}$ .

(2) Find the dot product of  $ \overrightarrow{m},\ \overrightarrow{n}$  denoted by  $ (\overrightarrow{m},\ \overrightarrow{n})$ .

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