
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 triangle $\triangle ABC$, $\overrightarrow{BA}=\overrightarrow{a}$, $\overrightarrow{BC}=\overrightarrow{b}$. If $O$ is its centroid, express $\overrightarrow{BO}$ in terms of $\overrightarrow{a}$ and $\overrightarrow{b}$ as ______; if $O$ is its circumcenter, and satisfies $\frac{AB}{BC}\overrightarrow{BC}\cdot\overrightarrow{BO}+\frac{BC}{AB}\overrightarrow{BA}\cdot\overrightarrow{BO}=2m{(\overrightarrow{BO})}^{2}$ ($m\in\mathbb{R}$), and $sinA+sinC=\sqrt{2}$, then the maximum value of $m$ is ______.

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