
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 functions $f(x)=x^{2}+mx+n^{2}$ and $g(x)=x^{2}+(m+2)x+n^{2}+m+1$, where $n\in\mathbb{R}$, find the maximum value of the real number $m$ such that for any $t\in\mathbb{R}$, at least one of $f(t)$ and $g(t)$ is non-negative.

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