
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 about $x$: $|x+1|\geqslant |x-2|+|t-3|$ has a solution. Find the maximum value $M$ of the real number $t$:
1. Find the maximum value $M$ of the real number $t$.
2. Under the condition of (1), given $a$, $b$, $c$ are positive numbers and $abc=2\sqrt{3}M$, find the minimum value of $\left(a+b\right)^{2}+c^{2}$.

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