
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
Let the real numbers $a_{1}, a_{2}, \cdots, a_{40}$ satisfy $a_{1} + a_{2} + \cdots + a_{40} = 0$, and for $1 \leqslant i \leqslant 40$, it holds that $\left|a_{i} - a_{i+1}\right| \leqslant 1$ (with $a_{41} = a_{1}$). Let $a = a_{10}$, $b = a_{20}$, $c = a_{30}$, and $d = a_{40}$. Find: (1) the maximum value of $a + b + c + d$; (2) the maximum value of $ab + cd$.

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