
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 triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and satisfy _____. Choose one of the two conditions from condition 1 and condition 2 to supplement the blank line above as known. 
$(1)$ Find angle $A$;
$(2)$ If $\triangle ABC$ is an acute triangle and $c=1$, find the range of values for the area of $\triangle ABC$.
Condition 1: $\left(b+c\right)\left(\sin B+\sin C\right)=a\sin A+3b\sin C$
Condition 2: $cos^2(\frac{π}{2}+A)+cosA=\frac{5}{4}$
Note: If conditions 1 and 2 are chosen to answer separately, the first answer will be scored.

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