
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 the acute triangle $\triangle ABC$, let $D$, $E$, and $F$ be the feet of the altitudes from $A$, $B$, and $C$ to the sides $BC$, $AC$, and $AB$ respectively. Then:

(1) $\frac{S_{\triangle AEF}}{S_{\triangle ABC}}=\cos^2 A, \frac{S_{BCEF}}{S_{\triangle ABC}}=\sin^2 A, \frac{S_{\triangle AEF}}{S_{BCEF}}=\cot^2 A$;

(2) $S_{\triangle DEF}=2S_{\triangle ABC} \cdot \cos A \cdot \cos B \cdot \cos C \leq \frac{1}{4}S_{\triangle ABC}$.

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