
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 Cartesian coordinate system, $O$ is the origin, and points $A$, $B$, $C$ are collinear, with point $O$ not on the line $AB$, satisfying $\overrightarrow{OC}=\frac{1}{3}\overrightarrow{OA}+λ\overrightarrow{OB}$.  
$(1)$ Find the value of $\lambda$;  
$(2) A(1,\cos x), B(1+\cos x,\cos x)$, $x∈[0,\frac{π}{2}]$, $f(x)=\overrightarrow{OA}•\overrightarrow{OC}-(2m+\frac{2}{3})|\overrightarrow{AB}|$, if the minimum value of $f(x)$ is $g(m)$, find the maximum value of $g(m)$.

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