
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 function $f(x) = ax^{3} + 3x^{2} + 1$, if there exist at least two real numbers $m$ such that $f(-m)$, $f(1)$, $f(m+2)$ form an arithmetic sequence, find the number of tangent lines that can be drawn to the curve $y=f(x)$ passing through the coordinate origin.

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