
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 \(ABCD\) given: \(M \in BC\) and \(BM:MC = 1:2\); \(N \in DC, DN:NC = 1:2\); \(\overline{AM} = \bar{a}\); \(\overline{AN} = \bar{b}\). Express the vectors \(\overline{AB}\), \(\overline{AD}\), \(\overline{MN}\), and \(\overline{BD}\) in terms of \(\bar{a}\) and \(\bar{b}\).

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