
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 $-1\leqslant a+b\leqslant 1$, $-1\leqslant a-b\leqslant 1$, find the range of $2a+3b$;  
If real numbers $a$, $b$, $c$ satisfy $a^{2}+b^{2}+c^{2}=6$, determine the relationship between $\frac{1}{a^{2}+1}+\frac{1}{b^{2}+2}$ and $\frac{1}{2}-\frac{1}{{c}^{2}+3}$ and explain the reason.

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