
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]


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Facial recognition technology is changing human life in various industries. Facial recognition refers to using computers to analyze facial videos or images, extract effective identification information from them, and ultimately determine the identity of the object. In facial recognition, distance testing is mainly used to detect the similarity between samples. Common ways to measure distance include Manhattan distance and cosine distance.

If there are two points $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$ in a two-dimensional space, then the Manhattan distance is defined as: $d(A,B) = |x_{1} - x_{2}| + |y_{1} - y_{2}|$. The cosine similarity is defined as: $\cos(A,B) = \frac{x_{1}}{\sqrt{x_{1}^{2} + y_{1}^{2}}} \times \frac{x_{2}}{\sqrt{x_{2}^{2} + y_{2}^{2}}} + \frac{y_{1}}{\sqrt{x_{1}^{2} + y_{1}^{2}}} \times \frac{y_{2}}{\sqrt{x_{2}^{2} + y_{2}^{2}}}$, and the cosine distance is $1 - \cos(A,B)$.

$(1)$ Given $A(-1,2)$ and $B(\frac{3}{5}, \frac{4}{5})$, find the Manhattan distance $d(A,B)$ and the cosine distance between $A$ and $B$.

$(2)$ Given $M(\sin \alpha, \cos \alpha)$, $N(\sin \beta, \cos \beta)$, and $Q(\sin \beta, -\cos \beta)$, if $\cos(M,N) = \frac{1}{5}$ and $\cos(M,Q) = \frac{2}{5}$, find the value of $\tan \alpha \tan \beta$.

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