
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 $f(x)=x^{3}+bx^{2}+cx+d$ whose graph passes through the point $P(0,2)$, and the equation of the tangent line at the point $M(-1,f(-1))$ is $6x-y+7=0$.

$(1)$ Find the expression for the function $y=f(x)$;

$(2)$ Determine the intervals of monotonicity for the function $y=f(x)$.

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