
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 $f(x) = x^3 + ax^2 + bx + 5$, the equation of the tangent line to the curve $y = f(x)$ at point $P(1, f(1))$ is $y = 3x + 1$.

(1) Find the values of $a$ and $b$;  
(2) Determine the intervals of monotonicity of $y = f(x)$ on $\mathbb{R}$;  
(3) Find the maximum value of $y = f(x)$ on the interval $[-3, 1]$.

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