
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 $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = x^2 + 2x$.
(I) Find the value of $f(0)$;
(II) Determine the analytic expression of the function on $\mathbb{R}$;
(III) If for any $t \in \mathbb{R}$, the inequality $f(t+1) + f(m-2t^2) < 0$ always holds, find the range of the real number $m$.

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