
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 vectors $a=(2\cos \alpha,\sin ^{2}\alpha)$ and $b=(2\sin \alpha,t)$, where $\alpha\in\left( 0,\frac{\pi}{2} \right)$ and $t$ is a real number.

$(1)$ If $a-b=\left( \frac{2}{5},0 \right)$, find the value of $t$;

$(2)$ If $t=1$ and $a\cdot b=1$, find the value of $\tan \left( 2\alpha+\frac{\pi}{4} \right)$.

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