
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 regular hexagon $ABCDEF$, points $P$, $Q$, $R$, and $S$ are chosen on sides $\overline{AB}$, $\overline{CD}$, $\overline{DE}$, and $\overline{FA}$ respectively, so that lines $PC$ and $RA$, as well as $QS$ and $EB$, are parallel. Moreover, the distances between these parallel lines are equal and constitute half of the altitude of the triangles formed by drawing diagonals from the vertices $B$ and $D$ to the opposite side. Calculate the ratio of the area of hexagon $APQRSC$ to the area of hexagon $ABCDEF$.

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