
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
On the edges \( AC, BC, BS, AS \) of a regular triangular pyramid \( SABC \) with vertex \( S \), points \( K, L, M, N \) are chosen respectively. It is known that points \( K, L, M, N \) lie in one plane, and \( KL = MN = 2 \), \( KN = LM = 9 \). In the quadrilateral \( KLMN \), there are two circles \(\Omega_{1}\) and \(\Omega_{2}\), where circle \(\Omega_{1}\) is tangent to the sides \( KN, MN \), and \( LM \), while circle \(\Omega_{2}\) is tangent to the sides \( KL, KN \), and \( LM \). The right circular cones \(\mathcal{F}_{1}\) and \(\mathcal{F}_{2}\) with bases \(\Omega_{1}\) and \(\Omega_{2}\) respectively are positioned inside the given pyramid, with the vertex \( P \) of cone \(\mathcal{F}_{1}\) on the edge \( AB \), and the vertex \( Q \) of cone \(\mathcal{F}_{2}\) on the edge \( CS \).

a) Find \(\angle SAB\).

b) Find the length of segment \( CQ \).

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