
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
In a factory, two different liquids were used to create two mixtures. For the first mixture, $a$ and $b$ grams of the liquids were used, respectively, and for the second mixture, $c$ and $d$ grams. The first mixture was $e$ times the amount of the second mixture. Both mixtures were tested in the laboratory by using $f$ grams from each. It was determined that the first mixture was usable because its density, although lower than the prescribed $g$ value, deviated by only $h$ per mille. Later, it was reported that the second mixture was not usable because its density was too high, deviating from $g$ by $i$ more per mille than the maximum allowable deviation. By then, $j$ grams of the first mixture had already been used. After mixing the remaining first mixture with the second mixture, it was found that the density of the new mixture deviated from $g$ by exactly the maximum allowable deviation. Express the maximum allowable per mille deviation and the densities of the original liquids in terms of the given data.

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