
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 $a \gt 0$, $b \gt 0$, and $\frac{1}{a}+\frac{2}{b}=2$.  
$(1)$ Find the minimum value of $2a+b$;  
$(2)$ If $4^{a}=3^{b}=t$, find the value of $t$.

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