
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
In the tetrahedron \( \mathrm{D}-\mathrm{ABC} \) shown, it is known that \( \mathrm{AB}=2 \) and \( \overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{BD}}=-3 \). Let \( \mathrm{AD}=a \), \( \mathrm{BC}=b \), and \( \mathrm{CD}=c \). Find the minimum value of \( \frac{\mathrm{c}^{2}}{\mathrm{ab}+1} \).

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