
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
A factory estimates that from the beginning of 2016, the total market demand $f(x)$ (in units) for a certain product over the first $x$ months has an approximate relation to the month number $x$ as follows: $f(x)=x(x+1)(35-2x)$, where $x \in \mathbb{N}^*$ and $x \leq 12$;  
(1) Write the relationship between the demand $g(x)$ for the product in the $x$-th month of 2016 and the month number $x$;  
(2) If the factory produces $a$ units of this product per month, what is the minimum value of $a$ to ensure that the market demand is met every month?

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