
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 two vectors in the plane, $\overrightarrow {a}=(\cos\alpha，(\lambda-1)\sin\alpha)$ and $\overrightarrow {b}=(\cos\beta，\sin\beta)$, where $\lambda>0$ and $0<\alpha<\beta<\frac{\pi}{2}$, if the vector $\overrightarrow {a}+ \overrightarrow {b}$ is perpendicular to $\overrightarrow {a}- \overrightarrow {b}$,  
(1) find the value of the real number $\lambda$;  
(2) if $\overrightarrow {a}\cdot \overrightarrow {b}= \frac{4}{5}$ and $\tan\alpha= \frac{4}{3}$, find the value of $\alpha$ (express the result in terms of inverse trigonometric functions).

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