
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 certain product has $m$ production sites, with respective outputs $a_{1}, a_{2}, \cdots, a_{m}$. There are $n$ sales sites, with respective demands $b_{1}, b_{2}, \cdots, b_{n}$. The unit transportation cost matrix for shipping from production sites to sales sites is given by
$$
\mathbf{C}=\left(c_{i j}\right)_{m \times n}
$$

where $c_{i j}$ is the unit transportation cost from production site $i$ to sales site $j$. Production and sales are balanced, meaning that
$$
\sum_{i=1}^{m} a_{m}=\sum_{j=1}^{n} b_{j}
$$

How should the product shipments be adjusted to minimize the total transportation cost?

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