
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 the function $f(x)=|x-1|+|x-t|$ $(t\in\mathbb{R})$

$(1)$ When $t=2$, find the solution set of the inequality $f(x) > 2$.

$(2)$ If for any $t\in[1,2],x\in[-1,3]$, $f(x)\geqslant a+x$ always holds, find the range of the real number $a$.

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