
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 that the numbers corresponding to points $A$, $M$, $N$, and $B$ on the number line are $-1$, $x-1$, $x+1$, and $11$ respectively. Segment $MN$ moves along the positive direction of the number line at a speed of $1$ unit per second. Let $t$ be the time of movement in seconds.  
$(1)$ Among all the segments formed by points $A$, $M$, $N$, and $B$, which segments have determinable lengths? Explain the reason.  
$(2)$ If $x=1$, answer the following two questions:  
① For what value of $t$ is $AM+BN=11$?  
② If points $A$, $B$, and segment $MN$ move simultaneously, point $A$ moves towards the positive direction of the number line at a speed of $2$ units per second, and point $B$ moves towards the negative direction at a speed of $1$ unit per second. During the movement, when $AM=BN$, find the value of $t$.

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