
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 inequality involving $x$: $|x-1|+|x+3| \leqslant m$ has a non-empty solution set, denote the minimum value of $m$ as $t$.

$(1)$ Find the value of $t$;

$(2)$ If positive real numbers $a$, $b$, and $c$ satisfy $a(2a+2c+b)=t-bc$, find the minimum value of $3a+b+c$.

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